Question

Show that 1+2^2+3^2+4^2+5^2+⋯.+n^2=(n(n+1)(2n+1))/6

Answer 1

us math induction prove method

Answer 2

if you dont know it ,so can you checking on wikipedia

Answer 3

I want the prove through concept of limit.

Answer 4

if you want to do it using a limit, you need the geometric formula. start of with:\[\frac{x^{n+1}-1}{x-1}=1+x+x^2+\cdots+x^n\] Take the derivative of both sides:\[\frac{d}{dx}\left(\frac{x^{n+1}-1}{x-1}\right)=1+2x+3x^2+\cdots+nx^{n-1}\]multiply by x:\[x\cdot\frac{d}{dx}\left(\frac{x^{n+1}-1}{x-1}\right)=x+2x^2+3x^3+\cdots+nx^n\]Take another derivative:\[\frac{d}{dx}\left(x\cdot\frac{d}{dx}\left(\frac{x^{n+1}-1}{x-1}\right)\right)=1+2^2x+3^2x^2+\cdots+n^2x^{n-1}\]I know this seems ridiculous, thats only because i didnt work out the details. You want to take the limit as x goes to 1 of both sides.

Answer 5

by the way, this is a very easy and standard proof by mathematical induction like jhonny said, is there a particular reason why you needed limits?

Answer 6

Yes I have a reason, unfortunately you got the answer.

Answer 7

"unfortunately"? im sry >.< i can delete it if need be.

Answer 8

No exactly, thanks.