calculating a power series

**buttonbear** · April 27th 2009, 03:14 PM

calculate a power series for $ln (1+x^4)$ , it's radius of convergence, and its interval of convergence

i'm really unsure of where to start/how to do this one, and i'm quite overwhelmed with these problems

please help- i wouldn't be asking so much, but this was assigned last minute and i really need it!

**skeeter** · April 27th 2009, 04:06 PM

Originally Posted by buttonbear

calculate a power series for $ln (1+x^4)$ , it's radius of convergence, and its interval of convergence

i'm really unsure of where to start/how to do this one, and i'm quite overwhelmed with these problems

please help- i wouldn't be asking so much, but this was assigned last minute and i really need it!

you should already know the power series for $\ln(1+x)$ ...

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

$\ln(1+x^3) = x^3 - \frac{x^6}{2} + \frac{x^9}{3} - \frac{x^{12}}{4} + ...$

determine the nth term and use the ratio test to find the radius and interval of convergence.

**buttonbear** · April 27th 2009, 05:34 PM

well, i wish i could say i learned it, but we're kind of left to struggle on our own...

so, for $ln (1+x^4)$ , would it be $= x^4 - \frac{x^8}{2} + \frac{x^12}{3} - \frac{x^{16}}{4} + ...$ ?

**skeeter** · April 27th 2009, 06:40 PM

Originally Posted by buttonbear

well, i wish i could say i learned it, but we're kind of left to struggle on our own...

so, for $ln (1+x^4)$ , would it be $= x^4 - \frac{x^8}{2} + \frac{x^12}{3} - \frac{x^{16}}{4} + ...$ ?

ok ... what's the nth term?

use the ratio test to find the interval of convergence ...

$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1$

**buttonbear** · April 27th 2009, 08:52 PM

the nth term is... $\frac{x^{4n}}{n}$ ?

edit: i mean.. $(-1^{n-1})\frac{x^{4n}}{n}$

editx2: actually..i'm not sure what the -1 is raised to, now that i think of it..

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