Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Compute the following limit $$\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{\sqrt{n^2+kn}}$$

Please I need your help asap



share|cite|improve this question

closed as off-topic by Najib Idrissi, Normal Human, drhab, jameselmore, TravisJ Sep 21 '15 at 12:47

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Najib Idrissi, Community, drhab, jameselmore, TravisJ
If this question can be reworded to fit the rules in the help center, please edit the question.

Any thoughts, efforts, ideas, insights...? – DonAntonio Nov 6 '12 at 19:00
$\lim_{x \to \infty} \frac{1}{\sqrt{n}} \sum{\frac{1}{\sqrt{n+k}}}$ – Matthew Nov 6 '12 at 19:04

$$\sum_{k=1}^n\frac{1}{\sqrt{n^2+kn}}=\frac{1}{n}\sum_{k=1}^n\frac{1}{\sqrt{1+\frac{k}{n}}}\xrightarrow [n\to\infty]{}\int_0^1\frac{dx}{\sqrt{1+x}}$$

share|cite|improve this answer
How did you get from step to step 3? – Matthew Nov 6 '12 at 19:10
If you mean the partition, it is $\,\{1/n\,,\,2/n\,,...,\,n/n=1\}\,$ of the unit interval $\,[0,1]\,$ – DonAntonio Nov 6 '12 at 19:11
@Matthew That step considers your sum as a Riemann Sum for the given integral. If this is something you're unfamiliar with, it's worth looking up. – Daniel Littlewood Nov 6 '12 at 21:29

Not the answer you're looking for? Browse other questions tagged or ask your own question.