Evaluate $1 + 2x + 3{x^2} + 4{x^3} + ........$ up to infinite, where $\left| x \right| < 1.$

Hint: It is given in the question that we have to Evaluate $1 + 2x + 3{x^2} + 4{x^3} + ........$ up to infinite, where $\left| x \right| < 1$ .
Let, $S = 1 + 2x + 3{x^2} + 4{x^3} + ........$ (I)
Then, multiply S by x.
Thus, subtract the equation of xS from the equation of S.
This will give a Geometric progression i.e. GP.
Hence, we will get the required answer on solving the equation further.

Complete step-by-step answer:
It is given in the question that we have to Evaluate $1 + 2x + 3{x^2} + 4{x^3} + ........$ up to infinite, where $\left| x \right| < 1$ .
Let, $S = 1 + 2x + 3{x^2} + 4{x^3} + ........$ (I)
Now, multiply S by x, we get,
 $\therefore xS = x\left( {1 + 2x + 3{x^2} + 4{x^3} + ........} \right)$
 $\therefore xS = x + 2{x^2} + 3{x^3}........$ (II)
Now, subtract equation (I) with equation (II), we get,
 \[\therefore \left( {S = 1 + 2x + 3{x^2} + 4{x^3} + ........} \right) - \left( {xS = x + 2{x^2} + 3{x^3}.......} \right)\] .
 \[\therefore S - xS = \left( {1 + 2x + 3{x^2} + 4{x^3} + ........} \right) - \left( {x + 2{x^2} + 3{x^3}.......} \right)\] .
\[\therefore S = \dfrac{1}{{{{\left( {1 - x} \right)}^2}}}\] \[\therefore S\left( {1 - x} \right) = 1 + x + {x^2} + {x^3} + ........\]

Note: Arithmetic Progression: An Arithmetic Progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. Difference here means second minus first. For instance, the sequence 5,7,9,11,13,15,17,…is an arithmetic progression with a common difference of 2.
General formula of Arithmetic Progression (AP) is ${a_n} = {a_m} + \left( {n - m} \right)d$
Geometric Progression: A geometric Progression, also known as geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio.
For example: the sequence 2,6,18,54,….is a geometric progression with common ratio 3.
General formula of Geometric Progression (GP) is $a,ar,a{r^2},a{r^3},a{r^4},.......$ where $r \ne 1$ is the common ratio and ‘a’ is a scalar factor.
Arithmetic-Geometric Progression: An Arithmetic-Geometric Progression (AGP) is a progression in which each term can be represented as the product of the terms of an Arithmetic Progression (AP) and Geometric Progression (GP).