lol i can truly say this is a coincidence: the reason why i started watching your videos was because i was struggling with this EXACT inequality which was assigned for my homework! Thank you very much for your help!
I think you have a mistake at 6:29, when you expand the fractions. Shouldn't it be 2 - 1/c + 1/(c+1)^2 = 2 + (-1)/c + 1/(c+1)^2 = 2 + ((-(c+1)^2) + c)/(c(c+1)^2) = 2 - ((c+1)^2 - c)/(c(c+1)^2) ?
For those saying this proof is incorrect, change the sign of c at 6:26 and work the proof out. The numerator will be c^2 + 2c + 1 - c which simplifies to c^2 + c + 1 = c(c+1) + 1. If you continue with the steps after you will get the same answer.
Induction Inequality Proof Example 1: Σ(k = 1 to n) 1/k² ≤ 2 - 1/n At time 5:44, how did you conjure up 2- (1/c+1) ? I mean, why do we have to specifically put it in that form? And how did you derive that expression?
***** Ah okay, I get it. I just wanna say, thanks for posting these videos. It helped me tremendously. I'm an undergrad in Comp Science and Discrete Math is such a torture.
It was originally - in front of the fraction and the sign inside the fraction was a +. You can see it as : x - (a + b) which can be written as x - a - b. With numbers : 6 - (3+2) = 6 - 5 = 1 but also = 6 - 3 - 2
Nice induction proof. Not a very strong inequality since we know summation(n=1 to infinity) 1/n^2 is famously equal to PI^2/6 = 1.64. compare to n=30 where 2-1/30 = 59/30 greater than PI^2/6
lol i can truly say this is a coincidence: the reason why i started watching your videos was because i was struggling with this EXACT inequality which was assigned for my homework! Thank you very much for your help!
Aajtac nuse
.
I think you have a mistake at 6:29, when you expand the fractions. Shouldn't it be 2 - 1/c + 1/(c+1)^2 = 2 + (-1)/c + 1/(c+1)^2 = 2 + ((-(c+1)^2) + c)/(c(c+1)^2) = 2 - ((c+1)^2 - c)/(c(c+1)^2) ?
Yeah, there is a mistake with the signs that he took. My answer and yours match , but the video is still good none the less...
This is the best lesson for induction I've seen even though it is just about a single example
For those saying this proof is incorrect, change the sign of c at 6:26 and work the proof out. The numerator will be c^2 + 2c + 1 - c which simplifies to c^2 + c + 1 = c(c+1) + 1. If you continue with the steps after you will get the same answer.
I wasn't really able to do this by using (c+1)² - c instead of (c+1)² + c at 6:30 . Just changes the solution .
Yeah, same, I've been stuck there for a while :/
At around 8:31 why are the terms subtracted instead?
All you did was split the fraction up, so why does it have to be subtracted?
***** Ahhh I see
Thanks Mr. Woo
Induction Inequality Proof Example 1: Σ(k = 1 to n) 1/k² ≤ 2 - 1/n
At time 5:44, how did you conjure up 2- (1/c+1) ? I mean, why do we have to specifically put it in that form? And how did you derive that expression?
***** Ah okay, I get it. I just wanna say, thanks for posting these videos. It helped me tremendously. I'm an undergrad in Comp Science and Discrete Math is such a torture.
How you make a + to - in this question can you please explain.
06:23, error, must be 2-\frac{(c+1)^2-c}{c(c+1)^2)}
At 8:43 , why did you change the fractions to subtraction instead of addition?
It was originally - in front of the fraction and the sign inside the fraction was a +. You can see it as : x - (a + b) which can be written as x - a - b. With numbers : 6 - (3+2) = 6 - 5 = 1 but also = 6 - 3 - 2
Thank you for your proof! I am trying to help one of my family members to understand proof by induction. Your method was easy to follow. Thanks again!
Thank you!! The algebra was so mind bending on this one. I gave up after a while. Thank you for the video!
Never give up! That's all.
It feels unreal, but *Never give up* !
This is really complicated! You can use a telescoping sum: Sum(k=1,n) 1/k^2
Thanks for the help! Better than my professor for sure!
Jesus Christ this man just saved my life
I like it second last line 3
6:28 is wrong with the signs
he's correct. he did not multiply the signs as he's just working on the fraction independently.
@@shujamukhtar4563 Which is something he cannot do, can he ?
I am stuck in a question of similar nature, could you please work it out for me..
1/1^1/2 + 1/2^1/2+...+ 1/n^1/2 > or equal to (n+1)^1/2
Why did you change the positive sign to negative sign ?? I think that is wrong !
The negative (from subtracting) has to apply all parts of the fraction. It is correct.
@@crawfordcarson134 Thank u so much, I was thinking that quarentine was blowing up my mind
prove that if x and y are natural numbers such that k=(x^2 +y^2)/xy is an integer, then k=2
Not going to use induction.
x²/xy +y²/xy=k
x/y + y/x =k
x|y and y|x only possible when x=y(for integers). Plug this in.
x/x +y/y = k
1+1=k
k=2😄
@@TechToppers Hahah... Thanks. But you are 4 years late bro.
Sorry but why its minus? when you change it from a plus to a minus you have to change < to > and than its wrong
Fynn X si
Fynn X si
When you multiplie, yes. He didn't use multiplication
In the 'Assume true for n=c' line, isn't it already given that c must be a positive integer, since c can only take the numbers that n can take?
Thank you for the wonderful explanation ! :)
Hii
Mistake with the sign at 6:29!! All wrong
Bro, its 10 PM and i need to send my activity at 11 PM and i get stuck on this fucking question. You are amazing men. Thanks men.
if c is negative, it still works.
Nice induction proof. Not a very strong inequality since we know summation(n=1 to infinity) 1/n^2 is famously equal to PI^2/6 = 1.64. compare to n=30 where 2-1/30 = 59/30 greater than PI^2/6
there is a symbol for positive integer: N
Thank you, this was very helpful.
This is beatiful!
by the way thanks ....
because of you I am clear with induction
This is beautiful
6:29 wrong
Prove that for all k ≤ n, pk(n) ≤ (n − k + 1)^k−1. Please someone solve it
i want the proof of:
1/2^2+1/3^2+1/4^2+...+1/n^2
I've another cool method without induction. Wanna know that instead??
thank u very much for making this vidd
the only good explanation there is for this type of inductions
Thank you so much! Really easy to follow :)
very nice explanation
thanks Yawei Hu
my hero :)
Thank you ❤
I love you
Thank YOU!
Thank you very much for the video, you saved my ass big time..
solve it again, it's wrong
ahhhhhhh founded 1st mistake in your video dude .....
jzt a small sign difference make that sum wholly different ......