Need help with Alternating Series Estimation Theorem for certain series.

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I am looking for some help with this series problem for calc 2.

Firstly I am to "test the following series for convergence or divergence."

$\sum_{n=1}^∞ \frac{(-1)^n}{n3^n}$

I have successfully managed to find that it converges, using the alternating series test for convergence. But now I am having an issue with the second task for this problem which is:

"If the series is convergent, use the Alternating Series Estimation Theorem to determine how many terms we need to add in order to find the sum with an error less than 0.0001? (If the quantity diverges, enter DIVERGES.)" _______ terms

I am getting stuck here because using the Alternating Series Estimation Theorem gives me an inequality that I don't know how to solve.

$b_{n+1}≤0.0001$

With $b_{n+1}$ being $\frac{1}{(n+1)3^{n+1}}$

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1 Answer

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You want to have $\frac1{(n+1)3^{n+1}}<\frac1{10^4}$, which is equivalent to $(n+1)3^{n+1}>10^4$. You can take $n=6$, because $3^7=2\,187$, and therefore $7\times3^7>5\times2\,000=10^4$.

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