Interesting Series Challenge!

Calculus Level 2

Find the sum of the following infinite series-

$\frac{1}{8}+\frac{1}{24}+\frac{1}{48}+\frac{1}{80}+\ldots$

Hint - Find a pattern between the numbers $\frac{1}{8},\frac{1}{24},\frac{1}{48}\ldots$

1/2

None of these

1/4

1/3

1/5

1 solution

Naren Bhandari
May 3, 2018

$\small { S = \dfrac{1}{8} + \dfrac{1}{24} + \dfrac{1}{48} + \dfrac{1}{80} +\dfrac{1}{120} \cdots = \dfrac{1}{8}\left(1 + \dfrac{1}{3} + \dfrac{1}{6} + \dfrac{1}{10} + \cdots\right) \\ S = \dfrac{1}{8}\left(1 + \dfrac{1}{1+2} + \dfrac{1}{1+2+3} + \dfrac{1}{1+2+3+4} + \dfrac{1}{1+2+\cdots +5 } \cdots\right) \\ S = \dfrac{1}{8} \sum_{n=1}^{\infty} \left(\dfrac{2}{n(n+1)}\right) = \dfrac{2}{8} \sum_{n=1}^{\infty}\left(\underbrace{\dfrac{1}{n} - \dfrac{1}{n+1}}_{\text{Telescoping sum}} \right) = \dfrac{1}{8}\times 2 =\dfrac{1}{4}}$

Interesting Series Challenge!

1 solution

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