Problem 56
Question
Find the sum of the series. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n} \pi^{2 n+1}}{3^{2 n+1}(2 n+1) !} $$
Step-by-Step Solution
Verified Answer
The sum of the series is \( \frac{\pi}{3 + \pi^2} \).
1Step 1: Identify the first term and the common ratio
The series is given by \( \sum_{n=0}^{\infty} \frac{(-1)^{n} \pi^{2 n+1}}{3^{2 n+1}(2 n+1) !} \). To identify the first term, substitute \( n = 0 \) into the series to get \( a = \frac{\pi}{3} \). The common ratio \( r \) is \( \frac{-\pi^2}{9} \).
2Step 2: Apply the formula for the sum of an infinite geometric series
The sum \( S \) of an infinite geometric series with first term \( a \) and common ratio \( r \) (where \( |r| < 1 \)) is given by the formula \( S = \frac{a}{1 - r} \).
3Step 3: Substitute and simplify
Substituting \( a = \frac{\pi}{3} \) and \( r = \frac{-\pi^2}{9} \) into the formula, we get \( S = \frac{\frac{\pi}{3}}{1 - \frac{-\pi^2}{9}} = \frac{\pi}{3 + \pi^2} \).
Other exercises in this chapter
Problem 56
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