An Introduction to Infinite Series

Helene
4 min readSep 10, 2021

In this article, we will cover the topic of infinite series. We will start out by showing the notation for an infinite series and after we will handle the subjects of divergence, convergence, and absolute convergence. If you are not already familiar with sequences and limits, then I recommend you to read these two articles first: ‘Finding Your Limits’ and ‘An Introduction to Sequences’.

The Notation

Let us first try to understand what is meant by an infinite series. To do so, we will first need to consider a sequence. Let us imagine that we have the following sequence:

This means that our sequence runs from one and until infinity. We can then say that we have:

Here, the s_n are called partial sums. We also notice that they form a sequence! The sequence can be written as:

So, what if we want to take the limit of this sequence of partial sums? This can be written as:

We can then say that

is our infinite series.

Convergence and Divergence of Infinite Series

One important thing to note is that if our sequence of partial sums, i.e.,

is convergent, then our infinite series is convergent as well. This also implies that we can say:

The same can be said for divergence — if the sequence of partial sums is divergent, then the infinite series is as well.

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