# Series Convergence Tests

Similar to the way there are multiple techniques of integration, we also learn multiple tests to determine whether or not a series converges or diverges.  Determining what convergence test to use can often be quite difficult when you learning calculus. In order to help my students practice this, I have made a Series Convergence Test Decision Tree. This is similar to the Techniques of Integration Decision Tree I made to help them when working with integrals.

I wanted to post this, in case you or your students would find such a resource helpful. If you would like to get some practice using it or testing series for convergence, I suggest trying to work through the following examples.

### Examples

1. \begin{align*}
\sum_{k=1}^{\infty}\frac{3}{3k-2}-\frac{3}{3k+1}
\end{align*}
2. \begin{align*}
\sum_{k=1}^{\infty}\frac{2^{k+2}}{3^k}
\end{align*}
3. \begin{align*}
\sum \frac{2k^{2}+1}{\sqrt{k^{3}+2}}
\end{align*}
4. \begin{align*}
\sum \left(\frac{k}{3k+3}\right)^{2k}
\end{align*}
5. \begin{align*}
\sum \frac{k!}{(2k)!}
\end{align*}
6. \begin{align*}
\sum \frac{(-1)^{k}}{k\ln(k)}
\end{align*}

## Conclusion

If you worked through the problems, let me know how you did.  Hopefully the decision tree proved to be helpful.  If it did, make sure to share it with anyone else that may find it useful.

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