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ISI2015-MMA-56

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Let $\{a_n\}$ be a sequence of non-negative real numbers such that the series $\Sigma_{n=1}^{\infty} a_n$ is convergent. If $p$ is a real number such that the series $\Sigma \frac{\sqrt{a_n}}{n^p}$ diverges, then

  1. $p$ must be strictly less than $\frac{1}{2}$
  2. $p$ must be strictly less than or equal to $\frac{1}{2}$
  3. $p$ must be strictly less than or equal to $1$ but can be greater than$\frac{1}{2}$
  4. $p$ must be strictly less than $1$ but can be greater than or equal to  $\frac{1}{2}$
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The numerator converges is given and now from p series we know for p<=1 series 1/n^p diverges . Now we use  the Cauchy-Schwarz inequality and the assumptions imply since we know the second sum converges, the third sum must diverge. 

$$\infty = \sum_{n=1}^\infty \frac{a_n^{1/2}}{n^p} \leq \left ( \sum_{n=1}^\infty a_n \right )^{1/2} \left ( \sum_{n=1}^\infty \frac{1}{n^{2p}} \right )^{1/2}.$$

 

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