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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Sequences and Series

Olympiad workbook · IOQM · Sequences and Series

If ∑_ k=1 ^N (2 k+1)/((k^2+k)^2)=0.9999 then determine the value of...

If

\sum_{k=1}^N \frac{2 k+1}{\left(k^2+k\right)^2}=0.9999 then determine the value of

N$.

Correct answer

99

Step-by-step solution

\begin{array}{r} \sum_{k=1}^N \frac{2 k+1}{\left(k^2+k\right)^2}=\sum_{k=1}^N \frac{2 k+1}{k^2(k+1)^2}=\sum_{k=1}^N \frac{(k+1)^2-k^2}{k^2(k+1)^2} \\ =\sum_{k=1}^N\left(\frac{1}{k^2}-\frac{1}{(k+1)^2}\right)=\left(\frac{1}{1^2}-\frac{1}{2^2}\right)+\left(\frac{1}{2^2}-\frac{1}{3^2}\right) \\ +\ldots+\left(\frac{1}{N^2}-\frac{1}{(N+1)^2}\right) \end{array}

\begin{aligned} & =1-\frac{1}{(N+1)^2}=\frac{9999}{10000} \\ & \Rightarrow \frac{1}{(N+1)^2}=\frac{1}{10000}=\frac{1}{(100)^2} \\ & \Rightarrow N+1=100 \\ & \Rightarrow N=99\end{aligned}

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