Find the radius and interval of convergence for the power series $\displaystyle{\sum_{k=1}^{\infty}} \frac{(x+3)^k}{k(6+(-1)^k)^k}$

I found that R=1 by calculating $\frac{1}{R} = \displaystyle{\limsup_{k->\infty}} |a_k|^{\frac{1}{k}}$. Also, since $c=-3$ we need to check the endpoints, $x=-2,-4$.

$S(-4)$ converges by the alternating series test $S(-2)$ has convergent geometric subseries and is therefore convergent.

I have a feeling this last step of checking the endpoints is incorrect because we don't know that it is geometric. Right?

  • $\begingroup$ The interval of convergence is (-8,2). Since your "solution" is far from this result, let me suggest that you explain in more details your approach, and in particular the specific theorems you rely on. $\endgroup$ – Did Mar 7 '14 at 7:32
  • $\begingroup$ I got it down to checking the endpoints, $-8$ and $2$. How would i determine if $S(2)$ and $S(-8)$ converge or diverge? $\endgroup$ – Skuttle_Butt Mar 7 '14 at 18:39
  • $\begingroup$ Quote: "in particular the specific theorems you rely on". $\endgroup$ – Did Mar 7 '14 at 20:17

Hint: Separate it into two subseries, for odd $k$'s and for even $k$'s.


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