# Find <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-TeXAtom-ORD">

Cierra Castillo 2022-07-08 Answered
Find
$\underset{n\to \mathrm{\infty }}{lim}{n}^{2}\sum _{k=1}^{n}\frac{1}{\left({n}^{2}+{k}^{2}\right)\left(\sqrt{{n}^{2}+{k}^{2}}+n\right)}.$
You can still ask an expert for help

• Live experts 24/7
• Questions are typically answered in as fast as 30 minutes
• Personalized clear answers

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Jaruckigh
For the last integral consider the indefinite integral
$I=\int \frac{1}{\sqrt{{x}^{2}+1}+1}dx=\int \frac{\sqrt{{x}^{2}+1}-1}{{x}^{2}}dx$
Here, you can use the hyperbolic trig substitution $x=\mathrm{sinh}t$ for the first summand in the numerator which yields

which after substituting back yields an antiderivative for I
$I={\mathrm{sinh}}^{-1}x-\frac{\sqrt{{x}^{2}+1}-1}{x}+C$
Note that this function is, as expected, regular at $x=0$, unlike the one in the 2nd equation above.