Series sum online calculator

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In order to calculate the series sum, one need simply make summation over all the elements of the series. For instance:

sum of the finity series

In the example above the summation procedure was very simple, as far as it was done the finity number of times. But what should we do if the upper summation bound is infinity? For instance, we need to find the sum of the following series:

example of the infinity series

As in the previous example, we can write this sum like this:

expanded form of infinity series

But what should we do next?! At this stage it is necessary to introduce the partial series sum conception. So, the partial series sum (denotes Sn) is called the sum of the first n terms of the series. I.e. in our case:

partial series sum

Knowing that, we can calculate the sum of the initial series as the limit of the partial series sum:

example of the infinity summation

Therefore, to calculate series sum, one needs somehow to find the expression of the partial series sum (Sn). In our case the series is the decreasing geometric progression with ratio 1/3. It is known that the sum of the first n elements of geometric progression can be calculated by the formula:

sum of the first n elements of the geometric series

where b1 - is the first element of the geometric series (in our case it equals to 1) and q - is the geometric series ratio (in our case 1/3). Therefore, the partial sum Sn for our series equals to:

example of partial series summation

Then the sum of our series (S) accoding the definition given above equals to:

example of the infinity summation

The examples above are very simple. Usually to calculate the series sum, one needs to make much more effort, and the main difficulty is to find the partial series sum. The online calculator below was created on the basis of the Wolfram Alpha, and it is able to find sum of highly complicated series. In addition, when the calculator fails to find series sum is the strong indication that this series is divergent (the calculator prints the message like "sum diverges"), so our calculator also indirectly helps to get information about series convergence.

To find sum of your series, you need to choose the series variable, lower and upper bounds and also input the expression for n-th term of the series.

Series sum calculator
Series to sum up
Input the expression which you want to sum up:

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Other useful links:

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