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So I was working on this problem

$$\sum\limits_{n=2}^\infty \frac{2^n}{n^{100}}$$

After all of my work performing the ratio test, I ended up with a $$\frac{2}{n+1} = 0$$

Therefore, I assumed it absolutely converges, but turns out that it diverges. May I please get an explanation why?

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I think you computed the ratio incorrectly. It should be $$\frac{2^{n+1}}{(n+1)^{100}} \cdot \frac{n^{100}}{2^n} = 2 \left(\frac{n}{n+1}\right)^{100}$$

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  • $\begingroup$ You're right, I did. I cancelled out the n in the numerator. Thanks for explaining. $\endgroup$ – Adan Vivero Apr 30 at 23:07
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A simple remark: you don't even need the ratio test, as this series diverges trivially: its general terù $\frac{2^n}{n^{100}}$ does not tend to $0$ at $\infty$.

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$$\frac{2^{n+1}}{(n+1)^{100}}\frac{n^{100}}{2^n}=\frac{2}{(1+\frac{1}{n})^{100}}$$

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