Sum of Series Calculator

Using the sum of series calculator, you can calculate the sum of an infinite series that has a geometric convergence as well as the partial sum of an arithmetic or geometric series. This summation solver can also help you calculate the convergence or divergence of a series.

Many a time, we'd want to calculate the sum of a series, and in order to do that, it helps to first know if the series is arithmetic or geometric. In an arithmetic series, the difference between each pair of successive terms is constant, while in a geometric series, the ratio between each pair of successive terms is constant.

For instance, let's consider the following series of the first 10 odd numbers:

$\footnotesize 1\! +\! 3\! +\! 5\! +\! 7\! +\! 9\! +\! 11\! +\! 13\! +\! 15\! +\! 17\! +\! 19$

This is an arithmetic series since the difference between any two successive pairs of numbers is 2. We can find the sum by using the following formula:

$S_n = \frac{n}{2}\ [2a + (n - 1) d]$

where:

• $n$ – Number of terms;
• $a$ – First term; and
• $d$ – Common difference.

We can also use the above formula to calculate the partial sum of an infinite arithmetic series. So, in the above example, the sum to 10 terms will be:

$S_{10} = \frac{10}{2}\ [2\times1 + (10 - 1)\times2]$

$S_{10} = 100$

If we have a geometric series, we will use a different formula to find the sum, which we'll take a look at below.

To know how to find the sum of a series in geometric progression, we can use either the finite sum formula or the infinite sum calculation. A geometric series can converge or diverge depending on the value of the common ratio $r$.

To decide on the convergence vs. divergence of a geometric series, we'd follow the following guideline based on the common ratio $r$:

• If $|r| > 1$, then the geometric series diverges and its sum to infinite terms can't be determined;

• If $|r| < 1$, then the geometric series converges to a finite sum and we can calculate the sum of the infinite series; and

• If $|r| = 1$, then the geometric series is periodic and its sum to infinite terms can't be determined.

On the other hand, to calculate the partial sum of a geometric series to a specific number of terms, we will use the formula:

$S_n = \frac{a_1\times (1-r^n)}{1-r}$

where:

• $a_1$ — First term;
• $r$ — Common ratio; and
• $n$ — Number of terms.

To calculate the sum of a series with a geometric convergence to an infinite number of terms, we will use the formula:

$S = \frac{a}{1 - r}$

where:

• $a$ — First term; and
• $r$ — Common ratio.

For example, consider the following geometric series:

$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$

Here, $a = 1$ and $r = \frac{1}{2}$.

So, the sum to an infinite number of terms is:

$S = \frac{1}{1 - \frac{1}{2}}$

which gives us:

$S = 2$

In this manner, we can calculate the sum of a geometric series with an infinite number of terms if the common ratio $r$ is between $-1$ and $1$.