Deep Rooted Tree

Algebra Level 4

Find the minimum value of non-negative integer $p$ such that $\displaystyle \sum_{k=1}^{\infty} \cfrac{1}{\sqrt{k(k+1)(k+2)...(k+p)}}$ converges.

You can try more of my fundamental problems here .

The series just doesn't converge. 0 3 2 1

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Brian Moehring
Jul 7, 2018

Relevant wiki: Convergence Tests

Note: For reasons that should become obvious, I'm going to use $q$ instead of $p$

For every non-negative integer $q$ , $\sum_{k=q+1}^\infty \frac{1}{k^{(q+1)/2}} = \sum_{k=1}^\infty \frac{1}{(k+q)^{(q+1)/2}} \leq \sum_{k=1}^\infty \frac{1}{\sqrt{k(k+1)(k+2)\cdots (k+q)}} \leq \sum_{k=1}^\infty \frac{1}{k^{(q+1)/2}}$

Therefore, the series in question is bounded on both sides by a $p$ -series with $p = \frac{q+1}{2}$ . It follows that it will converge if and only if $\frac{q+1}{2} = p > 1 \iff q > 1$

Since $q$ is an integer, the smallest value is $\boxed{2}$ .

@Brian Moehring Thanks for your solution. I did not realise there is a wiki page on convergence test in Brilliant.

donglin loo - 2 years, 11 months ago

Donglin Loo
Jul 7, 2018

Read this . It explains all.

Deep Rooted Tree

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