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8. Sequences and Series
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The sum to infinity of the following series $2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \frac{1}{{{2^3}}} + \frac{1}{{{3^3}}} + ........$, will be
A
$3$
B
$4$
C
$7/2$
D
$9/2$
Solution
(c) Given series
= $2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \frac{1}{{{2^3}}} + \frac{1}{{{3^3}}} + …….\infty $
$ = \left( {1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ……\infty } \right)$
+ $\left( {1 + \frac{1}{3} + \frac{1}{{{3^2}}} + \frac{1}{{{3^3}}} + ……\infty } \right)$
$ = \left( {\frac{1}{{1 – (1/2)}}} \right) + \left( {\frac{1}{{1 – (1/3)}}} \right) = 2 + \frac{3}{2} = \frac{7}{2}$.
Standard 11
Mathematics
