Problem 52 - Algebra and Trigonometry

Question

Find the partial fraction decomposition for \(\frac{2}{x(x+2)}\) and use the result to find the following sum: $$\frac{2}{1 \cdot 3}+\frac{2}{3 \cdot 5}+\frac{2}{5 \cdot 7}+\dots+\frac{2}{99 \cdot 101}$$

Step-by-Step Solution

Answer

The sum is \(\frac{100}{101}\)

Step 1: Partial Fraction Decomposition

Let the fraction \(\frac{2}{x(x+2)}\) be \(\frac{A}{x} + \frac{B}{x+2}\). Clear the denominators by multiplying the entire equation by \(x(x+2)\) to get \(2 = A(x+2) + Bx\). This forms a system of equations that can be solved to find the values of A and B.

Step 2: Solving the System of Equations

Substitute \(x=0\) into the equation to find \(A\) and substitute \(x=-2\) to find \(B\). This results in \(A = 1\) and \(B = 1\) respectively.

Step 3: Applying to the Series

Then, observe that the given series matches the pattern of the partial fraction decomposition. Thus, each term in the series can be written as \(\frac{1}{x(x+2)} = \frac{1}{x} - \frac{1}{x+2}\). Each pair of terms in the series will cancel out except the first fraction of the first term and the second fraction of the last term.

Step 4: Calculate the Sum

After the cancellations, the sum of the series becomes \(1 - \frac{1}{101}\) which equals \(\frac{100}{101}\)