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

kvkenyon
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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
 
Last edited:
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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.
 
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
 
well i got it...i ended up writing out cleaner when i posted it on here and then never realized the obvious...thanks
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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