Series

Real Analysis
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Tolaso J Kos
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Series

#1

Post by Tolaso J Kos »

Examine whether the following series converge or not. Justify your answer.
  1. \( \displaystyle \sum_{n=0}^{\infty}\left ( \frac{1}{n}-\frac{1}{2^n} \right ) \).
  2. \( \displaystyle \sum_{n=1}^{\infty}\left ( -1 \right )^n\cdot \frac{5n}{3n^2-2} \)
  3. \( \displaystyle \sum_{n=1}^{\infty}\frac{1}{\left ( n^{1/2014}+1 \right )\left ( n^{2/2014}+1 \right )\cdots \left ( n^{1007/2014}+1 \right )} \)
  4. \( \displaystyle \sum_{n=1}^{\infty}\left ( n^{1/2014}+1 \right )\left ( n^{2/2014}+1 \right )\cdots \left ( n^{1007/2014}+1 \right ) \)
  5. \( \displaystyle \sum_{n=0}^{\infty}\left ( \frac{1}{\sqrt{n+1}}-\frac{1}{5^n} \right ) \)
  6. \( \displaystyle \sum_{n=1}^{\infty}\left ( a_n \right )^n \) where \(a_n \) is a sequence of a zero limit.
Imagination is much more important than knowledge.
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