Math for Non-Geeks/Term test

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In this chapter we will see an easy and useful criterion for divergence. This criterion is called term test or also null-sequence test or divergence test. It says that every series $\sum _{k=1}^{\mathrm{\infty }}{a}_k$, where $(a_k{)}_{k\in \mathbb{N}}$ is not a null sequence, is divergent. When you invert that, this means that for every convergent series $\sum _{k=1}^{\mathrm{\infty }}{a}_k$ we have that$\underset{k\to \mathrm{\infty }}{\lim }{a}_k=0$( $(a_k{)}_{k\in \mathbb{N}}$ is a null sequence).

Example

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Outlook: Stronger version of term test

If we demand that $(a_n{)}_{n\in \mathbb{N}}$ is monotonically decreasing, then we can show that even $(n{a}_n{)}_{n\in \mathbb{N}}$ is a null sequence. See the respective exercise.

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