Series

The given series is: Step 1: Partial Fraction Decomposition
We express each term using partial fractions: Multiplying both sides by , we get: Expanding and equating coefficients: Solving for and : From the first equation, . Substituting in the second equation: Step 2: Telescoping the Series
Thus, rewriting the general term: Summing over 50 terms: This forms a telescoping series, where most terms cancel, leaving: Step 3: Evaluating the Final Expression
Thus, the sum of the series is:

The given series is: Step 1: Identifying the pattern Observing the general term: This is a standard series expansion for the function: From known mathematical results, the sum of the given infinite series converges to: Step 2: Conclusion Thus, the given series evaluates to: