Find the sum of infinite series 1^2.x^0 + 2^2.x^1 + 3^2.x^2 + 4^2.x^3 +…….

Given an infinite series and a value x, the task is to find its sum. Below is the infinite series 

1^2*x^0 + 2^2*x^1 + 3^2*x^2 + 4^2*x^3 +……. upto infinity, where x belongs to (-1, 1)

Examples:

Input: x = 0.5
Output: 12

Input: x = 0.9
Output: 1900

Approach:
Though the given series is not an Arithmetico-Geometric series, however, the differences and so on, forms an AP. So, we can use the Method of Differences.

Hence, the sum will be (1+x)/(1-x)^3.
Below is the implementation of above approach:

C++

// C++ implementation of above approach
#include <iostream>
#include <math.h>

using namespace std;

// Function to calculate sum
double solve_sum(double x)
{
    // Return sum
    return (1 + x) / pow(1 - x, 3);
}

// Driver code
int main()
{
    // declaration of value of x
    double x = 0.5;

    // Function call to calculate
    // the sum when x=0.5
    cout << solve_sum(x);

    return 0;
}

Java

// Java Program to find
//sum of the given infinite series
import java.util.*;

class solution
{
static double calculateSum(double x)
{

// Returning the final sum
return (1 + x) / Math.pow(1 - x, 3);

}

//Driver code
public static void main(String ar[])
{

  double x=0.5;
  System.out.println((int)calculateSum(x));

}
}
//This code is contributed by Surendra_Gangwar

Python

# Python implementation of above approach

# Function to calculate sum
def solve_sum(x):
    # Return sum
    return (1 + x)/pow(1-x, 3)

# driver code

# declaration of value of x
x = 0.5

# Function call to calculate the sum when x = 0.5
print(int(solve_sum(x)))

C#

// C# Program to find sum of
// the given infinite series
using System;

class GFG
{
static double calculateSum(double x)
{

// Returning the final sum
return (1 + x) / Math.Pow(1 - x, 3);

}

// Driver code
public static void Main()
{
    double x = 0.5;
    Console.WriteLine((int)calculateSum(x));
}
}

// This code is contributed
// by inder_verma..

PHP

<?php
// PHP implementation of
// above approach

// Function to calculate sum
function solve_sum($x)
{
    // Return sum
    return (1 + $x) /
            pow(1 - $x, 3);
}

// Driver code

// declaration of value of x
$x = 0.5;

// Function call to calculate
// the sum when x=0.5
echo solve_sum($x);

// This code is contributed
// by inder_verma
?>

Javascript

<script>
// javascript Program to find
//sum of the given infinite series


function calculateSum(x)
{

// Returning the final sum
return (1 + x) / Math.pow(1 - x, 3);

}

//Driver code

var x=0.5;
document.write(parseInt(calculateSum(x)));

// This code is contributed by 29AjayKumar

</script>

Output:

12

Time Complexity: O(1)

Auxiliary Space: O(1)