- #1
fauboca
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[tex]\sum_{n=2}^{\infty}z^n\log^2(n), \ \text{where} \ z\in\mathbb{C}[/tex]
[tex]\sum_{n=2}^{\infty}z^n\log^2(n) = \sum_{n=0}^{\infty}z^{n+2}\log^2(n+2)[/tex]
By the ratio test,
[tex]\lim_{n\to\infty}\left|\frac{z^{n+3}\log^2(n+3)}{z^{n+2}\log^2(n+2)}\right|[/tex]
[tex]\lim_{n\to\infty}\left|z\left(\frac{\log(n+3)}{ \log (n+2)}\right)^2\right| = |z|[/tex]
if [itex]|z|<1[/itex], then the sum converges, and if [itex]|z|>1[/itex], then the sum diverges.
Does this mean that [itex]R=1[/itex]?
[tex]\sum_{n=2}^{\infty}z^n\log^2(n) = \sum_{n=0}^{\infty}z^{n+2}\log^2(n+2)[/tex]
By the ratio test,
[tex]\lim_{n\to\infty}\left|\frac{z^{n+3}\log^2(n+3)}{z^{n+2}\log^2(n+2)}\right|[/tex]
[tex]\lim_{n\to\infty}\left|z\left(\frac{\log(n+3)}{ \log (n+2)}\right)^2\right| = |z|[/tex]
if [itex]|z|<1[/itex], then the sum converges, and if [itex]|z|>1[/itex], then the sum diverges.
Does this mean that [itex]R=1[/itex]?
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