Math Homework Help Please

[ndee]

Can't you do something with the Limes?

 

[chuckywang]

Originally posted by: MrCodeDude

Originally posted by: chuckywang

Originally posted by: MrCodeDude

Originally posted by: dighn

Originally posted by: MrCodeDude

Originally posted by: chuckywang
The answer is ln(5/4).

Note that 1-x+x^2-x^3+x^4 - x^5 + ... = 1/(1+x) by the infinite geomtric series for -1<x<1.
Taking integrals of both sides, we get:

x-x^2/2+x^3/3-x^4/4+x^5/5-x^6/x + ... = ln(1+x)
Note that when x = 0, left hand side = right hand side so we took care of the constant when we do indefinite integrals.

Your sum is precisely the case when x=1/4.

i.e. your sum is ln(1+1/4) = ln(5/4).

Okay, I understand where you got the 1/(1+x) sequence and how you integrated it to get the ln(x) sequence.

However, how do you know that x = 1/4?

if you expand the original series into the form x-x^2/2+x^3/3... you'll see that x is 1/4

eg if you expand it you have (1/4) - (1/4)^2/2 + (1/4)^3/3...

There is no X in the original sequence.

1) In my expansion of ln(1+x), plug in x=1/4.
2) Compare what you get with your sequence that you're trying to find the sum of.
3) Notice they are one and the same.
4) ...
5) Profit!

How did you find out that x = 1/4 in the first place though?

Experience, my dear Watson. I've seen that type of series before and I know how to sum it up. In mathematics, "how" is not important to proofs. You do not need to explain your moments of insights.