Serlo: EN: Overview: convergence criteria

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Decision tree for convergence and divergence of series

We already introduced a series $\sum _{k=1}^{\mathrm{\infty }}{a}_k$ as the sequence of the partial sums $S_n=\sum _{k=1}^n{a}_k$. A sequence is convergent, if the sequence of partial sums is convergent . Else the series is divergent. Assuming the series is convergent we define the value of the infinite sum of the series $\sum _{k=1}^{\mathrm{\infty }}{a}_k$ to be equal to the limit of the sequence.

In this chapter we will study different criteria or tests to determine whether a series is convergent or not. In further chapters, we will study each of this criteria more attentively and give a proof for each.

Criteria for convergence

We will give a proof for the following propositions in the respective main article for the criterion. Let a series $\sum _{k=1}^{\mathrm{\infty }}{a}_k$ be given. There is an arsenal of criteria to examine convergence:

Criteria for divergence

We are given a series $\sum _{k=1}^{\mathrm{\infty }}{a}_k$. To show that this series is divergent, there are multiple criteria:

Convergence is independent from starting index

In the section about the Cauchy test we saw that the starting index is irrelevant for the study of convergence. If we have a series of the form $\sum _{k=1}^{\mathrm{\infty }}{a}_k$, then we could also consider the series $\sum _{k=10}^{\mathrm{\infty }}{a}_k$ or $\sum _{k=4223}^{\mathrm{\infty }}{a}_k$. The only differences is the starting index $k$. This series all have the same convergence behaviour. So remember:

If we remove or alter finitely many summands, the individual values of the series will change of course, but the convergence behaviour stays the same. This fact is useful, you should always keep it in the back of your head. This could be useful in those cases, where you are not interested in the exact values of the series, but only if it converges or not.

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