Take this series: $$\sum_{n=0}^{\infty} \frac{nx^n}{n+9}.$$
I found the interval of convergence to be $-1 < x < 1$ but how do I approach finding the interval for which the series converges absolutely and conditionally?
I understand that for it to converge absolutely, each term put to the absolute should converge, otherwise it conditionally converges but I'm not sure where to start. Any tips?
A power series converges absolutely on the interior of the interval $(-R,R)$ where $R$ is the radius of convergence. For $|x|> R$ the series does not converge. Both of these claims follow from the proof of the cauchy hadamard formula. In your case for the boundary we are interested in the convergence of the series $$ \sum_{n=0}^\infty\frac{n}{n+9}\quad\text{and}\quad\sum_{n=0}^\infty\frac{n(-1)^n}{n+9} $$ But these series don't converge (why?).
Here are some general remarks, not tied to your particular example.The ratio test is guaranteed to fail at the end-points of the interval of convergence. At those points,more sophisticated methods are needed.My favourite is Raabe's test. Let $$\sum_{n=1}^{\infty}a_n$$ be a series of positive constants. Let $$\rho=\lim_{n \to \infty}(n(\frac{a_n}{a_{n+1}}-1)).$$ If $\rho>1,$ the sertes converges; if $\rho<1,$ the sertes diverges. Another useful technique is finding a partial sum by some method such as summation by parts and then taking a limit.
\sum_{n=0}^{\infty} \frac{nx^n}{n+9}.– projectilemotion Feb 24 '23 at 21:53