First slide

Special Series in Sequences and Series

Question

$The sum to n terms of the series$ $\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \dots \dots \dots \dots . . is$

Moderate

A

$n - 1 - 2^{- n}$
B

1
C

$n - 1 + 2^{- n}$
D

$1 + 2^{- n}$

Solution

$\begin{matrix} \frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \dots = (1 - \frac{1}{2}) + (1 - \frac{1}{4}) + (1 - \frac{1}{8}) + (1 - \frac{1}{16}) + \dots \\ = = n - \frac{\frac{1}{2} \{1 - \frac{1}{2^{n}}\}}{1 - \frac{1}{2}} \\ = n - 1 + 2^{- n} \end{matrix}$

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