First slide

Special Series in Sequences and Series

Question

If $H_{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$ then value of
$1 + \frac{3}{2} + \frac{5}{3} + \dots + \frac{2 n - 1}{n}$ is

Moderate

A

$H_{n} + n$
B

$2 n - H_{n}$
C

$n - 1 + H_{n}$
D

$H_{n} + 2 n$

Solution

$\begin{array}{l} 1 + \frac{3}{2} + \frac{5}{3} + \dots + \frac{2 n - 1}{n} \\ = (2 - 1) + (2 - \frac{1}{2}) + (2 - \frac{1}{3}) + \dots + (2 - \frac{1}{n}) \\ = 2 n - H_{n} \end{array}$

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