Series

The sum of a Sequences of terms.

Sigma notation

Given a function $f$ and two integers $A$ and $B$ (with $A\le B$), we write
${\sum }_{n=A}^{B}f(n)=f(A)+f(A+1)+\dots +f(B)$

i.e., adding up the results when $f$ is applied to each of the integers from $A$ to $B$.

$A$ is the starting point, the function is $f(n)$, and $B$ is the end point.

Sigma Notation Laws:

Note: There’s not necessarily a unique way of writing a sum using Sigma Notation.

Linearity of sigma notation

If $c$ is any constant (i.e., does not depend on $n$), then

$$\sum _{i=1}^{n}c=nc$$

and

$$\sum _{i=1}^{n}c{a}_i=c\sum _{i=1}^n{a}_i$$

and

$$\sum _{i=1}^n(a_i+b_i)$$

Splitting a Sum in Sigma Notation

If $m,n$ are integers with $1<m<n$, then

$$\sum _{i=1}^n{a}_i=\sum _{i=1}^m{a}_i+\sum _{i=m+1}^n{a}_i.$$

Sums of Powers (Used a lot - memorise them):

The sum of the first $n$ integers is given by:

$$\sum _{i=1}^{n}1=n$$

The sum of the first $n$ natural numbers is:

$$\sum _{i=1}^{n}i=\frac{1}{2}n(n+1)=\frac{n^2}{2}+\frac{n}{2}$$

The sum of the squares of the first $n$ natural numbers is:

$$\sum _{i=1}^n{i}^2=\frac{1}{6}n(n+1)(2n+1)=\frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{6}$$

The sum of the cubes of the first $n$ natural numbers is:

$$\sum _{i=1}^n{i}^3=\frac{1}{4}{n}^2(n+1{)}^2=\frac{n^4}{4}+\frac{n^3}{2}+\frac{n^2}{4}$$