First slide

Special Series in Sequences and Series

Question

The sum of the series $\frac{1}{1 + 1^{2} + 1^{4}} + \frac{2}{1 + 2^{2} + 2^{4}} + \frac{3}{1 + 3^{2} + 3^{4}} + \dots$ to n terms is

Moderate

A

$\frac{n (n^{2} + 1)}{n^{2} + n + 1}$
B

$\frac{n (n + 1)}{2 (n^{2} + n + 1)}$
C

$\frac{n (n^{2} - 1)}{2 (n^{2} + n + 1)}$
D

None of these

Solution

Given series is
$\frac{1}{1 + 1^{2} + 1^{4}} + \frac{2}{1 + 2^{2} + 2^{4}} + \frac{3}{1 + 3^{2} + 3^{4}} + \dots + n$ terms
Iet T₄be the nth term of the series
$\frac{1}{1 + 1^{2} + 1^{4}} + \frac{2}{1 + 2^{2} + 2^{4}} + \frac{3}{1 + 3^{2} + 3^{4}} + \dots$ $Then, T_{n} = \frac{n}{1 + n^{2} + n^{4}} = \frac{n}{{(1 + n^{2})}^{2} - n^{2}}$
$\begin{array}{l} = \frac{n}{(n^{2} + n + 1) (n^{2} - n + 1)} \\ = \frac{1}{2} [\frac{1}{n^{2} - n + 1} - \frac{1}{n^{2} + n + 1}] \\ = \frac{1}{2} [\frac{1}{1 + (n - 1) n} - \frac{1}{1 + n (n + 1)}] \end{array}$
$\begin{array}{l} ∴ T_{1} = \frac{1}{2} [\frac{1}{1} - \frac{1}{1 + 1 \cdot 2}] \\ T_{2} = \frac{1}{2} [\frac{1}{1 + 1 \cdot 2} - \frac{1}{1 + 2 \cdot 3}] \\ T_{3} = \frac{1}{2} [\frac{1}{1 + 2 \cdot 3} - \frac{1}{1 + 3 \cdot 4}] \\ \dots \dots \dots \\ \dots \dots \dots \\ T_{n} = \frac{1}{2} [\frac{1}{1 + (n - 1) n} - \frac{1}{1 + n (n + 1)}] \end{array}$
On adding all these equations, we get
$\sum_{r = 1}^{n} T_{r} = \frac{1}{2} [1 - \frac{1}{1 + n (n + 1)}] = \frac{n (n + 1)}{2 (n^{2} + n + 1)}$

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