Question

What is a series in math?

Answer 1

Series: A series is an operation of adding infinitely many terms in a sequence.

If $a_1,a_2,a_3,\cdots ,a_n,\cdots$ is a sequence, then the corresponding series is $a_1+a_2+a_3+\cdots +a_n+\cdots$

Types of series:

• Finite series: A series with a countable number of terms is called a finite series.

Example: $a_n=n(n+3)$ for $1\le n\le 6$

Then the series is: $\sum _{i=1}^6{a}_i=4+10+18+28+40+54$

$\Rightarrow$ $\sum _{i=1}^6{a}_i=154$

• Infinite series: A series with an infinite number of terms is called an infinite series.

Example: $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots \cdots$

This is a geometric series of the form $a_n=\frac{1}{2^n}$

Then the series is: $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots \cdots =\sum _{i=1}^{\infty }\frac{1}{2^n}=\frac{1}{1-{\displaystyle \frac{1}{2}}}=2\left[\mathbf{\because }\mathbf{sum}\mathbf{}\mathbf{of}\mathbf{}\mathbf{infinite}\mathbf{}\mathbf{GP}\mathbf{=}\frac{\mathbf{a}}{\mathbf{1}\mathbf{-}\mathbf{r}}\mathbf{,}\mathbf{a}\mathbf{=}\mathbf{first}\mathbf{}\mathbf{term}\mathbf{,}\mathbf{r}\mathbf{=}\mathbf{common}\mathbf{}\mathbf{ratio}\right]$