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Welcome to our Math lesson on The Special Types of Infinite Series, this is the fifth lesson of our suite of math lessons covering the topic of Infinite Series Explained, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
There are some special types of infinite series, which have many important applications in practice. Let's consider a few of them.
This is a special type of series which has the form
The p-series is special because it is convergent for p > 1 and divergent for 0 < p ≤ 1. For example, for p = 2, we obtain
Let's see what happens with the difference between two consecutive terms. Thus,
Since the difference becomes smaller and smaller, this is a converging series.
On the other hand, if p < 1, for example for p = 0.5, we obtain
It is true that the difference between terms decreases, but not so much so that to converge at a single point. In other words, this series is convergent to infinity, so practically it is divergent.
They represent a special case of the p-series where p = 1, i.e
In this way, we obtain for the harmonic series
This series is divergent, as it is impossible to reach to a limit point despite the decrease in the terms value with the increase of n.
This too, is a special series, which has the following form
We write the name of Euler Series (otherwise known as the Euler's Number) by e. It is worth to mention here that the counting of these series elements start from 0, not from 1. In this way, we obtain
This is a converging series, as the non-constant ratio between two consecutive terms is always smaller than 1, and moreover, it decreases more rapidly than the converging geometric series
we have discussed earlier in this tutorial (we saw that this series converges at 1). In fact, the Euler series gives an irrational number, the first digits of which are as follows
This number is very important in a wide number of topics both in math and in physics.
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