First slide

Special Series in Sequences and Series

Question

For a positive integer $n,$ let $a (n) = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{(2^{n}) - 1} .$ Then

Moderate

A

a (100) < 100
B

a (100) > 100
C

a (200) < 100
D

none of these

Solution

$\begin{array}{l} a (n) = 1 + (\frac{1}{2} + \frac{1}{3}) + (\frac{1}{4} + \dots + \frac{1}{7}) \\ + (\frac{1}{8} + \dots + \frac{1}{15}) + \dots + (\frac{1}{2^{n - 1}} + \frac{1}{2^{n - 1} + 1} + \dots + \frac{1}{(2^{n}) - 1}) \end{array}$
$< 1 + \frac{2}{2} + \frac{4}{4} + \frac{8}{8} + \dots + \frac{2^{n - 1}}{2^{n - 1}} = n$
Thus, a (100) < 100.

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