**Question 6: Finding the Exact Sum of the Series**Consider the series given by the equation below:\[ \sum_{n=1}^{\infty} \frac{1}{n(n+2)} \]The task is to find the exact sum of this infinite series.In the expression above:- \(\sum_{n=1}^{\infty}\) denotes the summation from \(n=1\) to infinity.- The term \(\frac{1}{n(n+2)}\) is the general term of the series.To solve this problem, one typically needs to use partial fraction decomposition or other series summation techniques.For educational purposes, we will guide you through the steps needed to find the exact sum of this series. Let's explore the solution together!**Step-by-step Solution:**1. **Partial Fraction Decomposition**: - Begin by decomposing the term \(\frac{1}{n(n+2)}\) into simpler fractions. - This can be expressed as: \[ \frac{1}{n(n+2)} = \frac{A}{n} + \frac{B}{n+2} \] - Solve for constants \(A\) and \(B\).2. **Finding Constants**: - Multiply both sides by \(n(n+2)\) to obtain a common denominator: \[ 1 = A(n+2) + Bn \] - Set up a system of equations to solve for \(A\) and \(B\): \[ 1 = An + 2A + Bn \] \[ 1 = (A + B)n + 2A \] - Equate coefficients of \(n\) and the constant term: \[ A + B = 0 \] \[ 2A = 1 \implies A = \frac{1}{2} \implies B = -\frac{1}{2} \]3. **Substituting Back**: - Substitute \(A\) and \(B\) back into the partial fractions: \[ \frac{1}{n(n+2)} = \frac{\frac{1}{2}}{n} + \frac{-\frac{1}{2}}{n+

Name: **Question 6: Finding the Exact Sum of the Series**Consider the serie
Uploaded: 2024-03-20T06:38:58.000Z
Channel: Bartleby
Description: **Question 6: Finding the Exact Sum of the Series**Consider the series given by the equation below:\[ \sum_{n=1}^{\infty} \frac{1}{n(n+2)} \]The task is to find the exact sum of this infinite series.In the expression above:- \(\sum_{n=1}^{\infty}\)