Power Series Representation

**kiddopop** · March 13th 2010, 09:21 AM

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?

**skeeter** · March 13th 2010, 10:35 AM

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 $|x|<1$ ...

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

integrate ...

$-\ln(1-x) = C + x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + ...<br />$

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

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

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

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

**kiddopop** · April 25th 2010, 01:04 PM

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

**skeeter** · April 25th 2010, 01:13 PM

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) ...

$\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...<br />$

... and proceed from there.

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

**kiddopop** · April 25th 2010, 01:19 PM

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

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