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For any integer $n\geqslant1$, how do you show that: $$\sum_{k=1}^{n} \frac{2^k}{\sqrt{k+0.5}}\leqslant2^{n+1}\sqrt{n+1}-\frac{4n^{\frac{3}{2}}}{3}$$

Do you need to find the formula of the partial sum of $\sum_{k=1}^{n} \frac{2^k}{\sqrt{k+0.5}}$ first before moving on to proof whether the inequality is true?

I could not find a suitable way of writing the formula of the partial sum, since this is a divergent series, or is there a way to do so?

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  • $\begingroup$ Have you tried induction? $\endgroup$
    – Aravind
    May 11, 2020 at 6:52
  • $\begingroup$ Have you tried integral comparison ? $\endgroup$
    – EDX
    May 11, 2020 at 7:02
  • $\begingroup$ Actually, this is supposed to be a high school contest math question... $\endgroup$
    – Cheng
    Dec 6, 2020 at 0:23

1 Answer 1

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Consider the following upper bound

$$\sum_{k=1}^n \frac{2^k}{\sqrt{k+1/2}} \leq \sum_{k=1}^n 2^k \leq 2^{n+1}.$$

This hints that the required inequality holds for sufficiently large $n$. We now show that it holds for all $n\geq 1$, i.e., for all $n\geq 1$ $$2^{n+1}\leq 2^{n+1}\sqrt{n+1}-4n^{1.5}/3.$$

First observe this holds for $n=1$. Then the claim follows since for all $n\geq 2$ $$\sqrt{n}\leq 2\cdot (\sqrt{n+1}-1) \quad \text{and}\quad 4\cdot n \leq 3\cdot 2^{n}.$$

The following manipulations prove the claim \begin{align} \sqrt{n}&\leq 2\cdot (\sqrt{n+1}-1)\\ 2^{n}\sqrt{n}&\leq 2^{n+1}\cdot (\sqrt{n+1}-1)\\ 4n^{1.5}/3&\leq 2^{n+1}\cdot (\sqrt{n+1}-1)\\ 2^{n+1}&\leq 2^{n+1}\sqrt{n+1}-4n^{1.5}/3. \end{align}

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