I just did a project using regression analysis with data from the last 30 year and estimated the MPC to be about .89-.86 depending upon the level of inequality in the country. In most text books you will see numbers at around .88-.91
Another corollary of this model is that the added effect on aggregate expenditures due to an increase in government spending is greater than the added effect on aggregate expenditures due to a decrease in total taxes. ΔΥ/ΔG = 1/(1-mpc) *>* ΔΥ/-ΔT = mpc/(1-mpc)
Math doesn't appear in fields such as physics, economics, engineering or even biology just because the people who included it want to feel smart. It appears out of necessity. As an example of how math simplifies things, try to describe something as simple as Newton's law of gravitation just qualitatively. It would take you at least a few lines to give all the details whereas the mathematical equation involves about 10 characters with giving the information in full generality.
Yt = total income. Yd = disposable income. Say mpc = b = 0.8. 1) Yt = C+I+NX+G 2) Yt = (1/(1-b)) (-bT+Co+I+NX+G). Yt = C+I+NX+G + $1. Yt increases $1, mpc is irrelevant. You can’t increment anything in parentheses in equation 2, because of the math order of operations. You must resolve parentheses, multiplication and division, then add the $1 to Yt. So Yt increases only $1, not $5. Yt = C+I+NX+G = (1/(1-b)) (-bT+Co+I+NX+G). Equations 1 and 2 must give the same answer, as equation 1.
My problem with Keynesianism: He thoroughly explains that Y = (1/mps)(b). So, in other words, to increase Y, with a fixed mps, we need to increase b. He listed a number of ways to increase b, one of which was increasing govt spending. My problem is that this assumes govt spending is productive and efficient. Empirical data says otherwise. Basically, nobody spends somebody else's money as carefully as they spend their own. Rather than how MUCH, we need to focus on how EFFICIENTLY we're spending.
Disagree. Gov spending works well where Market Failure exists due to positive and/or negative externalities. This is essentially why nations like the PRC and Singapore do stuff like build public housing - to account for positive externalities.
I agree with you at this point, but I do believe that government intervention is needed. The reason for this kind of thinking is the great depression we all saw what happens when the government didn't do anything Although I do believe their role should be minimized as changes in monetary policies can easily lead to recession or hyper inflation
It does not assume it is productive and efficient, although it is in market failures. The government spending is a private agents income, from that it follows a number o mechanisms in which economic activity can increase. This does not require any pareto efficiency assumption
If Yt = C+I+NX+G = 80+5+5+10 = 100, add $1 to the $10 G. G is then 11 and Yt is then 101. With Yt = (1/(1-b)) (-bT+a+I+NX+G) = (1/(1-0.8)) (-8+8+5+5+10) = 100, you can’t add $1 to G. You must follow the order of operations, add $1 to Yt, and again Yt = 101. Both equations must give the same answer as Yt = C+I+NX+G. If you added $1 to the $10 G inside the parentheses of the "multiplier" equation, you would get $105. So you are saying that $101 = $105. Three Stooges Math.
This doesn't make any sense!! Pandemic b= 2 trillion$ Propensity to save went up to 66.66% (2/3) If the formula was correct our GDP would increase thanks to the pandemic (by 3 trillion). I think Kayensian is missing a productivity factor. It's like the whole productivity rate was changed by the government.. No wonder why business people don't like him.
Kahn Academy ILLEGALLY puts addition first in the math order of operations. At 7:51 he says "if b gets shifted up" (an increment to b), then the impact on GDP will be whatever that shift (increment) is ... times the multiplier. That's nonsense. Y = (1/(1-c)) b You can't increment b, because it's already under the operation of multiplication. So the increment ... can't ... be ... multiplied. Say: Y = (1/(1-c)) b 10 = (5) 2 Add $1: 10 + 1 = (5) 2 + 1 10 + 1 = 10 + 1 11 = 11 Khan Academy says: 10 + 1 = (5) 2 + 1 10 + 1 = (5) 3 11 = 15 PEMDAS Parentheses, exponents, multiplication & division, addition & subtraction. Addition doe not come before multiplication.
+MrTugwit No, that is not what he is saying. This is the proper way to look at it: Y = 5(2) Y=10 Y=5(2+1) Y=15, and therefore the shift is the increment +1 times the coefficient, 5. We are not shifting the value of the entire function up 1 as you have written it, just b.
+Daniel Botkin Your "proper" way of looking at it: 5 x 2 = 10 5 x 2 + 1 = 15 is nonsense.PEMDAS says: 5 x 2 = 10 5 x 2 + 1 = 10 + 1 11 = 111) Y = C + I + NX + G Add $1 ΔG to G, and Y increases $1. ΔY = ΔG2) Y = (1/(1-c)) (-cT+Co+Ip+NX+G) You can't add $1 ΔG to G, in equation 2, because that would illegally put addition, before parentheses and multiplication, and give you a different result from equation 1. You have to resolve the entire right side of equation 2, and then add the $1 ΔG. So again ΔY = ΔG
+MrTugwit That's not what I wrote. 5(2+1) is the correct way to increment in this case. Like I said, we are not increasing the value of the function by 1 as you have stated. We are increasing Government Spending by 1, which would be contained in parentheses. Let's take a simple example. You have 4 cows who each produce 2 gallons of milk. That's a total of 8 gallons of milk. M = 2C; C=4 M= 2(4) = 8 Say we add an additional cow. How much milk can we produce? The correct way to write the expression is: M = 2(4+1) = 10. Given what we know about milk production per additional cow, it is incorrect to claim 5 cows would produce 9 gallons of milk, or in other words: 9 = 2(4) + 1. While this statement is mathematically true, it is not the correct one to analyze the problem. Do you see how this example is analogous to increasing gov spending by $1 when we have a multiplier? Each additional dollar gets multiplied, just like our milk production per cow.
+Daniel Botkin The Keynesian "multipliers" are not about variables, which represent completely different things, like cows and milk. All the variables are dollars, or ratios of dollars. What I wrote to start this off was taken from an equation in the video at about 5:58: Y = (1/(1-c)) b 10 = (5) 2 That says that $10 baseline income, is 5 times larger than it's $2 part: "b". Add $1 to the part of baseline income which is called "b": 10 + 1 = (5) 2 + 1 10 + 1 = 10 + 1 11 = 11 That's legal math. Why do you change the subject to cows and milk? when you can take the simple example from the video: 1) Y = C + I + NX + G 2) Y = (1/(1-c)) (-cT+Co+Ip+NX+G) In equation 1, you can directly increment G. In equation 2, you have to resolve the whole right side, before incrementing G. Otherwise you're doing illegal addition, before parentheses and multiplication. The Y and G in equation 2, are the same Y and G as in equation 1. Do you expect to get a different result with the 2 equations? You can get a different result ... with illegal math.
+MrTugwit Yes dollars are technically variables in the same sense, but that is besides the point. I used the example of milk and cows to explain functional relationships, but I can see now where your understanding falls apart. You claim the Y is the same in equations 1 and 2. Y is only the same in each equation if the economy is in equilibrium as was explained at 4:28. The first equation is an accounting identity used to represent total output. The second is the expenditures function you see shifting up and down the 45 degree line. Once the economy settles at equilibrium, that +1 you are adding to G will change hands through the economy, generating economic activity. If the mpc is .75, making the multiplier 4 [4 = (1/(1-.75)], then that plus 1 you added to G in equation 2 in time period 1 will actually result in the following change to equation 1 in time period 2: Y = C + 3 + I + NX + G + 1 You have to remember that time is key, because as you have treated the equations, the changes are happening instantaneously. Note that in the interim period, our point in expenditure and output space lies off the intersection between the 45 degree line and the expenditure function. An intuitive way to think about this is that one person's expenditure is another person's income. We can't think of an economy in terms of accounting identities only. A baker spends money on wheat, and the farmer spends money on farming equipment, and the manufacturers spend money on Ipods or whatever. The process continues, losing a little bit each time do to each actor's propensity to save. If the government increases spending (without a corresponding increase in taxes) $1 of income can go farther than just the $1 spent on the initial good or service -- and likewise, if the government lowers taxes (without a corresponding decrease in government spending). This could be financed from debt, although as +Jeremy Howson points out, in Keynes' model we could raise taxes and increase gov spending by the same amount and still generate economic activity (although this is economic activity in NOMINAL terms -- an important distinction). Is there anything I have not made clear?
You're talking nonsense. If you add $1 to G in [ E = Y = C(Y-T) +I+G+NX ] E = Y increases $1. The other equations which = Y, are derived from that 1st equation, and can be set equal to it. So they must give the same answer when $1 is added, that E increases only $1. E = Y cannot vary linearly AND geometrically with G, unless the "multiplier" is 1. What you're doing requires illegally putting addition first in the math order of operations.
The video is nonsense. At about 2:30, Khan Academy talks about -c1( - delta T), or c1(delta T). That's illegal subtraction: - delta T, before multiplication: -c1 x (-delta T), and illegal addition: delta T, before multiplication: c1 x (delta T) At about 7:15, he mentions the "infinite geometric series". An infinite series takes forever to complete. It would take forever for those lines to reach the 45 degree line. So there could never be an "equilibrium". At 9:55 he says: (delta Y) = 2.5 (delta b) That's illegal addition: delta b, before multiplication: 2.5 x (delta b). PEMDAS proves that he's talking nonsense.
+MrTugwit I have already pointed out in a lower comment the errors in your thinking regarding your perceived order of operations problem. As for the geometric series, please read up on convergent series. For example, the sum of the series .1^x will have a finite limit, but the functional value for each X as X approaches infinity will approach 0 very quickly, never quite reaching it. If we add up the terms, .1; .01; .001; .0001; etc, we end up with .1111....... and so on. Do you see how we're approaching some finite value to which further summations become more and more negligible? The same principle is in effect with the Keynesian multiplier.
+Daniel Botkin You've already pointed out that you don't know PEMDAS. You could write it P-E-MD-Deltas, because the deltas are addition or subtraction, and the come AFTER multiplication.The infinite series is about a supposed "multiplier" process which takes place in the real world, and which takes real time. Starting with an increment of spending, at each step, part is spent (c = mpc), and part is "saved". ΔY + cΔY + c^2ΔY + ... = (1/(1-c)) ΔG You approach the value 1/(1-c), but can never get there, because the "multiplier" process would take forever.And it's an infinite series of illegal addition: the deltas before multiplication. Khan Academy shows the "multiplier" lines extending beyond the 45 degree line.That's impossible. It would take longer than forever to get there.
+MrTugwit You are conflating several concepts here with the 45 degree line and infinite series. The point of the multiplier is that spending will increase, but the rate at which it increases will infinitely decay to $0/$1 of Y. While the process takes forever (in theory - in practice we can't get a value smaller than $.01), increases in spending become so small that they no longer add enough to substantially change the value of the multiplier. Say the multiplier is 1. It is good enough for our purposes to consider .999999999999999999....(continued to an infinite number of 9s) as the same thing.
+Daniel Botkin The Keynesian "multipliers" are supposed to show how much income, a spending increment will produce. The concepts are spending, income, an infinite series, time, and Keynesian equilibrium. That's the problem of the bozos who made up the story. With legal math, the infinite series, time, and Keynesian equilibrium are eliminated. The infinite series, which you have been avoiding is: ΔY + cΔY + c^2ΔY + ... = (1/(1-c)) ΔG And it's an infinite series of illegal addition before multiplication.
+MrTugwit Not avoiding. Say c = .25 and ΔY = $1. $1 + .25($1) + .0625($1) + .015625($1) + .... will not be a divergent series. Not sure what you were getting at by your original comment about "multiplier lines extending beyond the 45 degree line." Do you mean that the planned expenditure line would be forever shifting upwards because the multiplier is the sum of an infinite series? If that's the case you should take a calculus course and learn the concept of limits and divergent and convergent series. I'm sure there are plenty of great videos on Kahn Academy! See below for my response about "illegal addition"
I just did a project using regression analysis with data from the last 30 year and estimated the MPC to be about .89-.86 depending upon the level of inequality in the country. In most text books you will see numbers at around .88-.91
You should definitely go to khanacademy. org. It's incredibly easy to find videos there.
Watching whilst studying* :)
Thank you
2024 and it's still helpful :))
Another corollary of this model is that the added effect on aggregate expenditures due to an increase in government spending is greater than the added effect on aggregate expenditures due to a decrease in total taxes.
ΔΥ/ΔG = 1/(1-mpc) *>* ΔΥ/-ΔT = mpc/(1-mpc)
+Jeremy Howson
ΔΥ/ΔG = 1/(1-mpc)
ΔΥ = (1/(1-mpc)) ΔG
That's illegal addition: ΔG
before multiplication
this is great one. thanks
Math doesn't appear in fields such as physics, economics, engineering or even biology just because the people who included it want to feel smart. It appears out of necessity. As an example of how math simplifies things, try to describe something as simple as Newton's law of gravitation just qualitatively. It would take you at least a few lines to give all the details whereas the mathematical equation involves about 10 characters with giving the information in full generality.
Maybe you'd find the Khanacademy website easier to navigate?
Yt = total income. Yd = disposable income. Say mpc = b = 0.8.
1) Yt = C+I+NX+G
2) Yt = (1/(1-b)) (-bT+Co+I+NX+G).
Yt = C+I+NX+G + $1. Yt increases $1, mpc is irrelevant.
You can’t increment anything in parentheses in equation 2, because of the math order of operations. You must resolve parentheses, multiplication and division, then add the $1 to Yt. So Yt increases only $1, not $5.
Yt = C+I+NX+G = (1/(1-b)) (-bT+Co+I+NX+G).
Equations 1 and 2 must give the same answer, as equation 1.
Is the marginal propensity to consume (MPC) in the Keynesian model constant?
No, it depends on income
My problem with Keynesianism: He thoroughly explains that Y = (1/mps)(b). So, in other words, to increase Y, with a fixed mps, we need to increase b. He listed a number of ways to increase b, one of which was increasing govt spending. My problem is that this assumes govt spending is productive and efficient. Empirical data says otherwise. Basically, nobody spends somebody else's money as carefully as they spend their own. Rather than how MUCH, we need to focus on how EFFICIENTLY we're spending.
Disagree. Gov spending works well where Market Failure exists due to positive and/or negative externalities. This is essentially why nations like the PRC and Singapore do stuff like build public housing - to account for positive externalities.
I agree with you at this point, but I do believe that government intervention is needed. The reason for this kind of thinking is the great depression we all saw what happens when the government didn't do anything
Although I do believe their role should be minimized as changes in monetary policies can easily lead to recession or hyper inflation
It does not assume it is productive and efficient, although it is in market failures. The government spending is a private agents income, from that it follows a number o mechanisms in which economic activity can increase. This does not require any pareto efficiency assumption
isn't the y in c1y disposable income? so its diff from the aggregate income rite? so in tat case how do u factor out y wen they both mean diff things?
thanks for this :))
Math simplifies things
hi, its difficult to find your videos..they are just randomly located on you tube. i hope you can do some thing about this.
What is the official MPC in the US right now? Do economists just use an arbitrary number?
.8
If Yt = C+I+NX+G = 80+5+5+10 = 100, add $1 to the $10 G. G is then 11 and Yt is then 101.
With Yt = (1/(1-b)) (-bT+a+I+NX+G) = (1/(1-0.8)) (-8+8+5+5+10) = 100, you can’t add $1 to G.
You must follow the order of operations, add $1 to Yt, and again Yt = 101.
Both equations must give the same answer as Yt = C+I+NX+G.
If you added $1 to the $10 G inside the parentheses of the "multiplier" equation, you would get $105.
So you are saying that $101 = $105. Three Stooges Math.
W VIDEO
I'm Andrew
if you're comparing Keynes to von Hayek and Mises, please spell HAYEK properly. So who's the clown here now?
or he was insane
This doesn't make any sense!!
Pandemic b= 2 trillion$
Propensity to save went up to 66.66% (2/3)
If the formula was correct our GDP would increase thanks to the pandemic (by 3 trillion). I think Kayensian is missing a productivity factor. It's like the whole productivity rate was changed by the government.. No wonder why business people don't like him.
Kahn Academy ILLEGALLY puts addition first in the math order of operations.
At 7:51 he says "if b gets shifted up" (an increment to b),
then the impact on GDP will be
whatever that shift (increment) is ... times the multiplier. That's nonsense.
Y = (1/(1-c)) b
You can't increment b, because it's already under the operation of multiplication.
So the increment ... can't ... be ... multiplied.
Say:
Y = (1/(1-c)) b
10 = (5) 2
Add $1:
10 + 1 = (5) 2 + 1
10 + 1 = 10 + 1
11 = 11
Khan Academy says:
10 + 1 = (5) 2 + 1
10 + 1 = (5) 3
11 = 15
PEMDAS
Parentheses, exponents, multiplication & division, addition & subtraction.
Addition doe not come before multiplication.
+MrTugwit
No, that is not what he is saying.
This is the proper way to look at it:
Y = 5(2)
Y=10
Y=5(2+1)
Y=15, and therefore the shift is the increment +1 times the coefficient, 5.
We are not shifting the value of the entire function up 1 as you have written it, just b.
+Daniel Botkin Your "proper" way of looking at it:
5 x 2 = 10
5 x 2 + 1 = 15
is nonsense.PEMDAS says:
5 x 2 = 10
5 x 2 + 1 = 10 + 1
11 = 111) Y = C + I + NX + G
Add $1 ΔG to G, and Y increases $1.
ΔY = ΔG2) Y = (1/(1-c)) (-cT+Co+Ip+NX+G)
You can't add $1 ΔG to G, in equation 2,
because that would illegally put addition, before parentheses and multiplication,
and give you a different result from equation 1.
You have to resolve the entire right side of equation 2,
and then add the $1 ΔG.
So again ΔY = ΔG
+MrTugwit
That's not what I wrote.
5(2+1) is the correct way to increment in this case. Like I said, we are not increasing the value of the function by 1 as you have stated. We are increasing Government Spending by 1, which would be contained in parentheses.
Let's take a simple example. You have 4 cows who each produce 2 gallons of milk. That's a total of 8 gallons of milk.
M = 2C; C=4
M= 2(4) = 8
Say we add an additional cow. How much milk can we produce?
The correct way to write the expression is:
M = 2(4+1) = 10.
Given what we know about milk production per additional cow, it is incorrect to claim 5 cows would produce 9 gallons of milk, or in other words: 9 = 2(4) + 1. While this statement is mathematically true, it is not the correct one to analyze the problem. Do you see how this example is analogous to increasing gov spending by $1 when we have a multiplier? Each additional dollar gets multiplied, just like our milk production per cow.
+Daniel Botkin
The Keynesian "multipliers" are not about variables, which represent completely different things, like cows and milk.
All the variables are dollars, or ratios of dollars.
What I wrote to start this off was taken from an equation in the video at about 5:58:
Y = (1/(1-c)) b
10 = (5) 2
That says that $10 baseline income, is 5 times larger than it's $2 part: "b".
Add $1 to the part of baseline income which is called "b":
10 + 1 = (5) 2 + 1
10 + 1 = 10 + 1
11 = 11
That's legal math.
Why do you change the subject to cows and milk?
when you can take the simple example from the video:
1) Y = C + I + NX + G
2) Y = (1/(1-c)) (-cT+Co+Ip+NX+G)
In equation 1, you can directly increment G.
In equation 2, you have to resolve the whole right side, before incrementing G.
Otherwise you're doing illegal addition, before parentheses and multiplication.
The Y and G in equation 2, are the same Y and G as in equation 1.
Do you expect to get a different result with the 2 equations?
You can get a different result ... with illegal math.
+MrTugwit
Yes dollars are technically variables in the same sense, but that is besides the point.
I used the example of milk and cows to explain functional relationships, but I can see now where your understanding falls apart.
You claim the Y is the same in equations 1 and 2.
Y is only the same in each equation if the economy is in equilibrium as was explained at 4:28.
The first equation is an accounting identity used to represent total output.
The second is the expenditures function you see shifting up and down the 45 degree line.
Once the economy settles at equilibrium, that +1 you are adding to G will change hands through the economy, generating economic activity. If the mpc is .75, making the multiplier 4 [4 = (1/(1-.75)], then that plus 1 you added to G in equation 2 in time period 1 will actually result in the following change to equation 1 in time period 2:
Y = C + 3 + I + NX + G + 1
You have to remember that time is key, because as you have treated the equations, the changes are happening instantaneously. Note that in the interim period, our point in expenditure and output space lies off the intersection between the 45 degree line and the expenditure function.
An intuitive way to think about this is that one person's expenditure is another person's income. We can't think of an economy in terms of accounting identities only. A baker spends money on wheat, and the farmer spends money on farming equipment, and the manufacturers spend money on Ipods or whatever. The process continues, losing a little bit each time do to each actor's propensity to save. If the government increases spending (without a corresponding increase in taxes) $1 of income can go farther than just the $1 spent on the initial good or service -- and likewise, if the government lowers taxes (without a corresponding decrease in government spending). This could be financed from debt, although as +Jeremy Howson points out, in Keynes' model we could raise taxes and increase gov spending by the same amount and still generate economic activity (although this is economic activity in NOMINAL terms -- an important distinction).
Is there anything I have not made clear?
lol 99.9% of the clowns who say things like this are not even qualified to make such claims
ok that was just ignorant. you don't really believe that
You're talking nonsense. If you add $1 to G in [ E = Y = C(Y-T) +I+G+NX ] E = Y increases $1. The other equations which = Y, are derived from that 1st equation, and can be set equal to it. So they must give the same answer when $1 is added, that E increases only $1. E = Y cannot vary linearly AND geometrically with G, unless the "multiplier" is 1. What you're doing requires illegally putting addition first in the math order of operations.
I hate keynes economics!
I think it's a misnomer to say that one is a "Keynesian" when he/she stands for stimulating the economy by lowering taxes. That idea predates Keynes.
Marry me.
The video is nonsense.
At about 2:30, Khan Academy talks about -c1( - delta T), or c1(delta T).
That's illegal subtraction: - delta T,
before multiplication: -c1 x (-delta T),
and illegal addition: delta T,
before multiplication: c1 x (delta T)
At about 7:15, he mentions the "infinite geometric series".
An infinite series takes forever to complete.
It would take forever for those lines to reach the 45 degree line.
So there could never be an "equilibrium".
At 9:55 he says:
(delta Y) = 2.5 (delta b)
That's illegal addition: delta b, before multiplication: 2.5 x (delta b).
PEMDAS proves that he's talking nonsense.
+MrTugwit
I have already pointed out in a lower comment the errors in your thinking regarding your perceived order of operations problem.
As for the geometric series, please read up on convergent series. For example, the sum of the series .1^x will have a finite limit, but the functional value for each X as X approaches infinity will approach 0 very quickly, never quite reaching it. If we add up the terms, .1; .01; .001; .0001; etc, we end up with .1111....... and so on. Do you see how we're approaching some finite value to which further summations become more and more negligible? The same principle is in effect with the Keynesian multiplier.
+Daniel Botkin You've already pointed out that you don't know PEMDAS.
You could write it P-E-MD-Deltas,
because the deltas are addition or subtraction,
and the come AFTER multiplication.The infinite series is about a supposed "multiplier" process
which takes place in the real world,
and which takes real time. Starting with an increment of spending,
at each step, part is spent (c = mpc), and part is "saved".
ΔY + cΔY + c^2ΔY + ... = (1/(1-c)) ΔG
You approach the value 1/(1-c), but can never get there,
because the "multiplier" process would take forever.And it's an infinite series of illegal addition: the deltas
before multiplication. Khan Academy shows the "multiplier" lines extending beyond the 45 degree line.That's impossible. It would take longer than forever to get there.
+MrTugwit
You are conflating several concepts here with the 45 degree line and infinite series.
The point of the multiplier is that spending will increase, but the rate at which it increases will infinitely decay to $0/$1 of Y. While the process takes forever (in theory - in practice we can't get a value smaller than $.01), increases in spending become so small that they no longer add enough to substantially change the value of the multiplier. Say the multiplier is 1. It is good enough for our purposes to consider .999999999999999999....(continued to an infinite number of 9s) as the same thing.
+Daniel Botkin
The Keynesian "multipliers" are supposed to show how much income, a spending increment will produce.
The concepts are spending, income, an infinite series, time, and Keynesian equilibrium.
That's the problem of the bozos who made up the story.
With legal math, the infinite series, time, and Keynesian equilibrium are eliminated.
The infinite series, which you have been avoiding is:
ΔY + cΔY + c^2ΔY + ... = (1/(1-c)) ΔG
And it's an infinite series of illegal addition before multiplication.
+MrTugwit
Not avoiding. Say c = .25 and ΔY = $1.
$1 + .25($1) + .0625($1) + .015625($1) + .... will not be a divergent series.
Not sure what you were getting at by your original comment about "multiplier lines extending beyond the 45 degree line." Do you mean that the planned expenditure line would be forever shifting upwards because the multiplier is the sum of an infinite series? If that's the case you should take a calculus course and learn the concept of limits and divergent and convergent series. I'm sure there are plenty of great videos on Kahn Academy!
See below for my response about "illegal addition"