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Special Series in Sequences and Series

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Question

$The sum of the first 10 -terms of the series \frac{5}{1.2.3} + \frac{7}{2.3.9} + \frac{9}{3.4.27} + \dots is$

Moderate

Solution

$T_{n} = \frac{n^{t h} term of the series}{n (n + 1) \cdot 3^{n}} = \frac{2 n + 3}{n (n + 1) 3^{n}}$
$\begin{array}{l} Let \frac{2 n + 3}{n (n + 1) 3^{n}} = (\frac{3}{n} - \frac{1}{n + 1}) \frac{1}{3^{n}} \\ = \frac{1}{n 3^{n - 1}} - \frac{1}{3^{n} (n + 1)} \\ S_{n} = \sum T_{n} = \sum [\frac{1}{n 3^{n - 1}} - \frac{1}{3^{n} (n + 1)}] = (1 - \frac{1}{2.3}) + (\frac{1}{2.3} - \frac{1}{{3.3}^{2}}) + . . + [\frac{1}{n \cdot 3^{n} - 1} - \frac{1}{(n + 1) 3^{n}}] \\ = [1 - \frac{1}{(n + 1) 3^{n}}] \\ S_{10} = 1 - \frac{1}{{11.3}^{10}} \end{array}$

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