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prove that:

$$\displaystyle\sum_{k=n^2+1}^{n^2+2n+1}\sqrt{k}\le 2n^2+2n+\dfrac{7}{6},n\ge 1$$

I find this inequality is very strong. Thank you!

such:when $n=100$,we use the mathmatic $$\displaystyle\sum_{k=10001}^{10201}\approx 20201.16666254125412794937\cdots $$

and $2n^2+2n+\dfrac{7}{6}|_{n=100}=20201.166666666666\cdots$

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The sum has $2n+1$ terms each of which is no greater than $\sqrt{n^2+2n+1}=n+1$. The sum can thus be bounded by $(2n+1)(n+1)=2n^2+3n+1$ which is a little weaker than what you are asking for. –  Eckhard Mar 28 '13 at 15:02
    
Thank you ,Are sure? you have $2n^2+3n+1>2n^2+2n+\dfrac{7}{6}$!,but is wrong –  math110 Mar 28 '13 at 15:16
    
Do you know integration? –  Calvin Lin Mar 28 '13 at 15:18
    
@math110 why do you think that what I wrote is wrong? –  Eckhard Mar 28 '13 at 15:21

2 Answers 2

up vote 1 down vote accepted

Note that your bound gets much tighter as $n$ increases, suggesting that some kind of limiting argument is involved.

With that insight, we have that

$$ \begin{align} \sum_{k = n^2 + 1}^{k=n^2+2n+1} \sqrt{k} & = \sum_{k=n^2} ^{k=n^2 + 2n} \sqrt{k} + 1 \\ & = 1 + \sum_{k=n^2} ^{k=n^2+2n} \int_k^{k+1} \sqrt{k} \, dx \\ & \leq 1 + \sum_{k=n^2} ^ {k=n^2 + 2n} \int_{k}^{k+1} \sqrt{x} \, dx \\ & = 1 + \int_{n^2} ^{(n+1)^2} \sqrt{x} \, dx\\ & = 1 + [ \frac {2}{3} k^{\frac {3}{2}} ]_{n^2} ^{(n+1)^2} \\ & = 1 + [\frac {2}{3} (3n^2 + 3n+1) ] \\ & = 2n^2 + 2n + \frac {5}{3} \end{align} $$

Now, let's consider how much we over approximated by. Looking at the inequality, we over approximated by $\int_{k}^{k+1} \sqrt{x} - \sqrt{k}$. Since the square root function is concave, we have the extra triangle on top, which tells us that we over approximated by at least $\frac {1}{2} (\sqrt{k+1}-\sqrt{k})$. Summing this up, we get $$\sum_{k=n^2}^{n^2+2n} \frac {1}{2} ( \sqrt{k+1}- \sqrt{k} ) = \frac {1}{2} ( k+1 - k) = \frac {1}{2}. $$

Hence, (back)accounting for this over-approximation, we know that

$$ \sum_{k = n^2 + 1}^{k=n^2+2n+1} \sqrt{k} \leq 2n^2 + 2n + \frac {5}{3} - \frac {1}{2} = 2n^2 + 2n + \frac {7}{6}. $$

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Since $x\mapsto\sqrt{x}$ $\ (x>0)$ is a concave function we have for any $m>0$ the inequality $${1\over 2}\bigl(\sqrt{m}+\sqrt{m+1}\bigr)\leq\int_m^{m+1}\sqrt{x}\ dx\ .$$ Putting $$m:=n^2+k\qquad (0\leq k\leq 2n)$$ and summing over $k$ we get $${1\over2}\sqrt{n^2} +\sum_{k=1}^{2n}\sqrt{n^2 +k}+{1\over2}\sqrt{n^2+2n+1} \leq \int_{n^2}^{(n+1)^2}\sqrt{x}\ dx={2\over3}\bigl((n+1)^3-n^3\bigr)\ .$$ Now extend the sum on the LHS to $k=2n+1$ and collect the remains. You get $$\sum_{k=1}^{2n+1}\sqrt{n^2 +k}-{1\over2}\leq 2n^3+2n+{2\over3}\ ,$$ which is equivalent with the stated inequality.

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