Finding Closed Form Expressions for Telescoping Sums: A Practical Guide
Telescoping sums are a fascinating area of mathematics, offering a succinct and elegant way to solve complex series. One common method involves using partial fractions and cancellation principles to simplify the sum. In this article, we will explore how to find a closed form expression for the specific telescoping sum ( sum_{k1}^{n} frac{1}{k^2} - frac{1}{(k 1)^2} ). We will break down the process step-by-step and provide a practical example using the computational engine WolframAlpha.What is a Telescoping Sum?
A telescoping sum is a series where most of the terms cancel out, leaving only a few terms to be summed. This simplifies the calculation significantly. The given sum is a classic example of a telescoping sum.Given the sum ( S sum_{k1}^{n} frac{2(k 1)}{k^2(k 1)^2} ), our goal is to find a closed form expression for ( S ).
Step-by-Step Solution
Step 1: Simplify the General Term
First, let's simplify the general term of the sum. We start by rewriting the term using partial fractions. [ frac{2(k 1)}{k^2(k 1)^2} frac{k(k 1) (k 1)}{k^2(k 1)^2} frac{k}{k^2(k 1)^2} frac{1}{k^2(k 1)^2} ] This can be further simplified to: [ frac{2(k 1)}{k^2(k 1)^2} frac{2}{k(k 1)^2} ] Next, we decompose the fraction into partial fractions. [ frac{2}{k(k 1)^2} frac{A}{k} frac{B}{(k 1)} frac{C}{(k 1)^2} ] Multiplying both sides by ( k(k 1)^2 ) to clear the denominators, we get: [ 2 A(k 1)^2 Bk(k 1) Ck ] Expanding and equating coefficients, we find that ( A 1 ), ( B -1 ), and ( C 0 ). Therefore, the term simplifies to: [ frac{2}{k(k 1)^2} frac{1}{k} - frac{1}{(k 1)} ]Step 2: Write the Sum
Now we can write the sum as follows: [ S sum_{k1}^{n} left( frac{1}{k} - frac{1}{k 1} right) ] This is a telescoping series, where most terms will cancel out. Specifically, we have: [ S left( frac{1}{1} - frac{1}{2} right) left( frac{1}{2} - frac{1}{3} right) left( frac{1}{3} - frac{1}{4} right) ldots left( frac{1}{n} - frac{1}{n 1} right) ] All intermediate terms cancel, leaving us with: [ S 1 - frac{1}{n 1} ] Therefore, the closed form expression for the sum is: [ S_n 1 - frac{1}{(n 1)^2} ]Verification Using Computational Tools
To verify this solution, we can use WolframAlpha. Plugging the expression into the search bar, we get the same result.WolframAlpha: Making the World’s Knowledge Computable