-1
$\begingroup$

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

Please I need your help asap

Cheers

Matthew

  • 4
    Any thoughts, efforts, ideas, insights...?2012-11-06
  • 0
    $\lim_{x \to \infty} \frac{1}{\sqrt{n}} \sum{\frac{1}{\sqrt{n+k}}}$2012-11-06

1 Answers 1

2

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

  • 0
    How did you get from step to step 3?2012-11-06
  • 1
    If you mean the partition, it is $\,\{1/n\,,\,2/n\,,...,\,n/n=1\}\,$ of the unit interval $\,[0,1]\,$2012-11-06
  • 0
    @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.2012-11-06