1. calculating a power series

calculate a power series for $\displaystyle ln (1+x^4)$, 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 problemsplease help- i wouldn't be asking so much, but this was assigned last minute and i really need it!

2. Originally Posted by buttonbear
calculate a power series for $\displaystyle ln (1+x^4)$, 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 problemsplease 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 $\displaystyle \ln(1+x)$ ...

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

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

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

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

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

4. 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 $\displaystyle ln (1+x^4)$, would it be $\displaystyle = x^4 - \frac{x^8}{2} + \frac{x^12}{3} - \frac{x^{16}}{4} + ...$?
ok ... what's the nth term?

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

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

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

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

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