Question

Find the sum of each series in Exercises 45–52.∑ (from n = 1 to ∞) [ (2n + 1) / (n²(n + 1)²) ]

Accepted Answer

Start by examining the general term of the series: $$\frac{2n + 1}{n$$^{2}$$(n + 1$$)^{2}$$}. $$Our goal is to express this term in a form that allows us to use telescoping or known series sums. Try to decompose the term into partial fractions. Since the denominator is $$n^{2}(n$$ + $$1)^{2}$$, consider expressing the term as a sum of fractions of the form $$\frac{A}{n} + \frac{B}{n$$^{2}$$} + \frac{C}{n + 1} + \frac{D}{(n + 1$$)^{2}$$}. $$Set up the equation: $$\frac{2n + 1}{n$$^{2}$$(n + 1$$)^{2}$$} = \frac{A}{n} + \frac{B}{n$$^{2}$$} + \frac{C}{n + 1} + \frac{D}{(n + 1$$)^{2}$$}. $$Multiply both sides by $$n^{2}(n$$ + $$1)^{2} to $$clear denominators and solve for constants $$A$$, $$B$$, $$C$$, and $$D by $$equating coefficients of powers of $$n. $$Once the partial fractions are found, rewrite the series as a sum of simpler series: $$\sum_{n=1}^{\infty} \left( \frac{A}{n} + \frac{B}{n$$^{2}$$} + \frac{C}{n + 1} + \frac{D}{(n + 1$$)^{2}$$} \right). $$This will allow you to separate the original series into sums that are easier to evaluate or telescope. Evaluate the sums by recognizing telescoping patterns or using known series results such as the Riemann zeta function for $$\sum \frac{1}{n$$^{2}$$}. $$Combine the results carefully to find the sum of the original series.

Find the sum of each series in Exercises 45–52.
∑ (from n = 1 to ∞) [ (2n + 1) / (n²(n + 1)²) ]

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Key Concepts

Infinite Series and Convergence

Partial Fraction Decomposition

Telescoping Series