# calculating a power series

• April 27th 2009, 04:14 PM
buttonbear
calculating a power series
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(Doh)please help- i wouldn't be asking so much, but this was assigned last minute and i really need it!
• April 27th 2009, 05:06 PM
skeeter
Quote:

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(Doh)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.
• April 27th 2009, 06:34 PM
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 $= x^4 - \frac{x^8}{2} + \frac{x^12}{3} - \frac{x^{16}}{4} + ...$?
• April 27th 2009, 07:40 PM
skeeter
Quote:

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 ...

$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1$
• April 27th 2009, 09:52 PM
buttonbear
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..