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Let we have the following sequence $$y_n= \frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\ \sqrt{n^2+n}}$$ find the limit of the sequence $y_n$ decide whether it is increasing or decreasing

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    Username: "Arab belly punching"? – 2017-01-08
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    @parsiad Something that gives us the laughs but none of our business. – 2017-01-08

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Hint
Prove that $$\frac{n}{\sqrt{n^2+n}}\leq \sum\frac{1}{\sqrt{n^2+k}}\leq\frac{n}{\sqrt{n^2}}$$ $$\frac{n}{\sqrt{n^2+n}}\leq y_n\leq\frac{n}{\sqrt{n^2}}$$ Then use squeeze lemma
It is a increasing sequence
Knowing that $$\frac{n}{\sqrt{n^2+n}}\leq y_n\leq\frac{1}{n}$$ We have that $y_n\leq\frac{1}{n}$ and $\frac{n+1}{\sqrt{(n+1)^2+n+1}}\leq y_{n+1}$
And easy you can show that $$\frac{1}{n}\leq \frac{n+1}{\sqrt{(n+1)^2+n+1}}$$ So $$y_ n\leq y_{n+1}$$

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    Can you explain it more ? – 2017-01-08
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    Increasing or decreasing sequence ? – 2017-01-08
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    @Arabbellypunching The sum is larger than $n$ times the smallest number the sum reaches. Likewise, the sum is smaller than than $n$ times the largest number the sum reaches. – 2017-01-08
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    How can I prove it is increasing or decreasing sequence ? – 2017-01-08
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    @SimpleArt how can I prove whether this sequence is increasing or decreasing ? – 2017-01-09
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    @Arabbellypunching I edit the answer. – 2017-01-09
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    This is strange: this answer was upvoted, never downvoted, accepted even, but the argument about the monotonicity is completely wrong. – 2017-03-05
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    The claim following "Knowing this" contains a mistake, since $\frac{n}{\sqrt{n^2}}$ simplifies to $1$, not $\frac{1}{n}$. – 2017-03-06