1. ## Power Series Representation

Find a power series representation for the function f(x)=ln(1-x^2). Write the series in sigma notation. For which values of x is the power series representation of f valid?

2. Originally Posted by kiddopop
Find a power series representation for the function f(x)=ln(1-x^2). Write the series in sigma notation. For which values of x is the power series representation of f valid?
for $\displaystyle |x|<1$ ...

$\displaystyle \frac{1}{1-x} = 1 + x + x^2 + x^3 + ...$

integrate ...

$\displaystyle -\ln(1-x) = C + x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + ...$

for $\displaystyle x = 0$ , $\displaystyle C = 0$ ...

$\displaystyle \ln(1-x) = -\left(x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + ... \right)$

$\displaystyle \ln(1-x^2) = -\left(x^2 + \frac{x^4}{2} + \frac{x^6}{3} + \frac{x^8}{4} + ... \right)$

I'll leave you to find the sigma notation for the series.

3. ## Power Series For an Indefinite Integral

Find a power series for the indefinite integral of cos(x^2). Use both sigma notation and expanded form.

4. Originally Posted by kiddopop
Find a power series for the indefinite integral of cos(x^2). Use both sigma notation and expanded form.
start w/ the series for cos(x) ...

$\displaystyle \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...$

... and proceed from there.

btw, next time ... start a new problem w/ a new thread.

5. Ok. Thank you. And I thought I had. Oops.