Power Series Convergence from Terms

Question

My homework wants me to find the interval of convergence and the radius of the power series, but it does not give the function, instead it gives the first 4 terms of the series.

The terms are $-x^{9} + \frac{x^{11}}{8} - \frac{x^{13}}{27} + \frac{x^{15}}{64}$

I am confused on how to accomplish this.

Answer 1

Well you should see some regularities which enable you to "guess" all the other terms. It is not mathematically strict, but hey, it's your homework:

1. First, you should see that your series is alternating
2. Next, notice that the power of $x$ is always increased by 2
3. Finally notice that denominator is always $\frac{1}{n^3}$

So your series is $$ \sum_{n=1}^{\infty} (-1)^n \frac{x^{7+2n}}{n^3} $$