First slide

Special Series in Sequences and Series

Question

$The sum to infinity of the series \frac{1}{1} + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + \dots \dots \dots \dots \dots \dots is equal to:$

Moderate

A

2
B

$\frac{5}{2}$
C

3
D

none of these

Solution

$\begin{array}{l} S = \frac{1}{1} + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + \dots \dots \dots \dots . \\ t_{n} = \frac{1}{1 + 2 + 3 + \dots \dots . . + n} = \frac{2}{n (n + 1)} = 2 (\frac{1}{n} - \frac{1}{n + 1}) \\ t_{1} = 2 (1 - \frac{1}{2}) \\ t_{2} = 2 (\frac{1}{2} - \frac{1}{3}) \\ t_{3} = 2 (\frac{1}{3} - \frac{1}{4}) and so on \\ t_{n} = 2 (\frac{1}{n} - \frac{1}{n + 1}) \\ S_{n} = 2 (1 - \frac{1}{n + 1}) \end{array}$
$S_{\infty} = 2$

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