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