Power series representation of ln((1+2t)/(1-2t))

[kvkenyon]

Homework Statement

Find a power series representation for the function $$f(t) = \ln((1+2t)/(1-2t))$$

Homework Equations

$$f(t) = \ln((1+2t)/(1-2t))$$

The Attempt at a Solution

$$\ln(1+2t)-\ln(1-2t)$$

take derivative of f(t) expanded

$$\frac{2}{1+2t}+\frac{2}{1-2t}$$

$$2 \int \frac {1}{1-(-2t)} + 2 \int \frac{1}{1-2t}$$

$$2 \int \displaystyle\sum_{n=o}^{\infty} -2^n x^n + 2 \int \displaystyle\sum_{n=0}^{\infty} 2^n x^n$$

$$2 \int 1 - 2x + 4x^2 - 8x^3 + 16x^4 +... + 2 \int 1 + 2x + 4x^2 + 8x^3 +16x^4 +32x^5+...$$

then i combine them left with $$2 \int 2 + 8x^2 + 32x^4 + ...$$


then i get stuck

 

[Dick]

I'm not sure why you switched from t to x. But then I'm not sure why you are stuck either. Just integrate term by term.

 

[kvkenyon]

it was an accident I am used to typing x and i didnt realize. also the answer choices that i have are with n-1 all over and i usually end up with n+1

 

[kvkenyon]

well i got it...i ended up writing out cleaner when i posted it on here and then never realized the obvious...thanks