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In a former article, we were introduced to infinite series. We were also briefly introduced to the concept of absolute and conditional convergence, but we lacked some tools before we could look at examples. In this article, we will look at two of the concepts we needed: Geometric and Harmonic Series.
Geometric Series
We will start by looking at Geometric Series. Let us first introduce its general notation, which can be denoted in two manners:
These two series are identical, they are just written differently. Therefore, they will also have the same value in case of convergence.
As we saw in our former article, we need to know the partial sums if we want to take the limit of a series. So, let us see them:
We then also remember that we needed to take the limit of these partial sums. The result, whether they diverge or converge, is the same for our series. I will not show the whole process, but we end up with:
We can then clearly see that a geometric series will converge if we have: |r|<1, or otherwise written as: -1 < r < 1.
Geometric Series Limit Examples
We have now seen how to find the limit of a geometric series. Let us now take a few examples, where we check whether a series diverges or converges.
Imagine that we have the following series, which we want to check whether diverges or converges:
