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# Sum Calculator

## Sum Calculator compute the sum and convergence of the series. To start working with the calculator, specify an indexed variable, index of summation, lower and upper bound of summation following the rules for entering numbers and functions.

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### Summation

Summation in mathematics is denoted by a capital letter sigma Σ.

#### Summation examples

Let's sum the natural numbers from 1 to 10, then m = 1, n = 10.

$\sum _{i=1}^{10}i$ = $\left(1\right)$$+$$\left(2\right)$$+$$\left(3\right)$$+$$\left(4\right)$$+$$\left(5\right)$$+$$\left(6\right)$$+$$\left(7\right)$$+$$\left(8\right)$$+$$\left(9\right)$$+$$\left(10\right)$ = $55$

Calculate the sum of fractions, where an indexed variable = $\frac{n}{2}$, lower bound of summation = 1, upper bound of summation = 5. If the summation index can be any variable, then instead of i we use the index n in this example.

$\sum _{n=1}^{5}\left(\frac{n}{2}\right)$ = $\left(\frac{1}{2}\right)$$+$$\left(\frac{2}{2}\right)$$+$$\left(\frac{3}{2}\right)$$+$$\left(\frac{4}{2}\right)$$+$$\left(\frac{5}{2}\right)$ = $\frac{15}{2}$ = $7.5$

Now let's write as an indexed variable an expression that, in addition to the summation index, contains other variables. Compute the sum, where an indexed variable = $\frac{n}{m+2}$, lower bound of summation = 1, upper bound of summation = 4.

$\sum _{n=1}^{4}\left(\frac{n}{m+2}\right)$ = $\left(\frac{1}{m+2}\right)$$+$$\left(\frac{2}{m+2}\right)$$+$$\left(\frac{3}{m+2}\right)$$+$$\left(\frac{4}{m+2}\right)$ = $\frac{10}{m+2}$

Let's take another example. Compute the sum, where an indexed variable = $\frac{2-n}{n+3}$, lower bound of summation = -2, upper bound of summation = 3.

$\sum _{n=-2}^{3}\left(\frac{2-n}{n+3}\right)$ = $\left(\frac{2-\left(-2\right)}{-2+3}\right)$$+$$\left(\frac{2-\left(-1\right)}{-1+3}\right)$$+$$\left(\frac{2-\left(0\right)}{0+3}\right)$$+$$\left(\frac{2-\left(1\right)}{1+3}\right)$$+$$\left(\frac{2-\left(2\right)}{2+3}\right)$$+$$\left(\frac{2-\left(3\right)}{3+3}\right)$ = $\frac{25}{4}$ = $6.25$
##### Infinite Series

The sum of an infinite number of terms, also called the sum of the series - is a mathematical expression with which you can write an infinite number of terms.

If the limit of the terms of the sum is equal to a finite number, then such a series is called convergent.

If the limit of the terms of the sum does not exist, or it is equal to infinity, then such a series diverges and is called divergent.

For example, the harmonic series is divergent:

$\sum _{n=1}^{\infty }\left(\frac{1}{n}\right)$ = $\infty$
The series $\sum _{n=1}^{\infty }\left(\frac{1}{n}\right)$is divergent