Hello Tedszy, Nice video very much appreciated. When I saw the definition there were 2 things I thought about : 1) proving that the harmonic sum diverges using Theorem of Lagrange 2) it might just be the difference between the sum of rectangles of size 1/k * 1 when k=1..n and the area under the graph of 1/x from taken 1 to n I am somewhat fond of Theorem of Lagrange or theorem of finite growth as it is very useful in understanding calculus with the property that f'(x)>0 function is increasing as a consequence of this. Well theorem of Lagrange states that given f(x):[a,b]->I with f continuous on [a,b] and derivable on (a,b) then there will exist between a and b at least a point c where the tangent to the graph is parallel to the line passing trough points (a,f(a)) and (b,f(b)) or equivalent a1/(k+1) k=1 1> ln(2) -ln(1)> 1/2 k=2 1/2>ln(3)-ln(2)>1/3 ....... k=n 1/n>ln(n+1)-ln(n)>1/(n+1) adding up sum(1/i ; i=1..n)>ln(n+1)-ln(1) doing a limit as n goes to infinity shows that the harmonic series diverges When I was 11th grade learning this theorem and Th of Cauchy and applications there wasn't anything such as this in my school years that make you feel both smart and dumb at the same time. Smart because you see the graphical explanation and dumb because you can't see where you need to manipulate stuff to make the problems work
That's a good question. I'll have to look into it. AFAIK, Euler discovered all this himself. Possibly "Mascheroni" is added to disambiguate from Euler's other constant, e.
Hello Tedszy,
Nice video very much appreciated. When I saw the definition there were 2 things I thought about :
1) proving that the harmonic sum diverges using Theorem of Lagrange
2) it might just be the difference between the sum of rectangles of size 1/k * 1 when k=1..n and the area under the graph of 1/x from taken 1 to n
I am somewhat fond of Theorem of Lagrange or theorem of finite growth as it is very useful in understanding calculus with the property that f'(x)>0 function is increasing as a consequence of this.
Well theorem of Lagrange states that given f(x):[a,b]->I with f continuous on [a,b] and derivable on (a,b) then there will exist between a and b at least a point c where the tangent to the graph is parallel to the line passing trough points (a,f(a)) and (b,f(b)) or equivalent
a1/(k+1)
k=1 1> ln(2) -ln(1)> 1/2
k=2 1/2>ln(3)-ln(2)>1/3
.......
k=n 1/n>ln(n+1)-ln(n)>1/(n+1)
adding up sum(1/i ; i=1..n)>ln(n+1)-ln(1)
doing a limit as n goes to infinity shows that the harmonic series diverges
When I was 11th grade learning this theorem and Th of Cauchy and applications there wasn't anything such as this in my school years that make you feel both smart and dumb at the same time. Smart because you see the graphical explanation and dumb because you can't see where you need to manipulate stuff to make the problems work
Thanks fod the comments and merry Christmas! I will think about your use of Lagrange's theorem over the holidays!
this is hypnotizing
We will need these introductory ideas for the next couple of vids.
great topic!
Thanks!
Papa Flammy of "Flammable Maths" calls it the "Oily-Macaroni Constant".
Calculus with pasta
Finally. I've been just using it as a magical number lol
Most of us have heard of Euler. I know very little of Mascheroni. What part did he play in discovering the Euler Mascheroni constant?
That's a good question. I'll have to look into it. AFAIK, Euler discovered all this himself. Possibly "Mascheroni" is added to disambiguate from Euler's other constant, e.
Like, comment, Subscribe! Follow me on FB: facebook.com/profile.php?id=61559517069850
∞ - ∞ = about 0.6.
LOL
genius