Question

Evaluate


3n

Lim ∑ rootn/(n+3r)^3/2

n→∞ r=0

Answer 1

$lim_{n\to \infty}\sum \frac{\sqrt{n}}{(n+3r)^{3/2}} , r=0$

$lim_{n\to \infty}\sum \frac{\sqrt{n}}{(n)^{3/2}}=lim_{n\to \infty}\sum \frac{1}{n}$

$1/n>0$ and decreasing with n, then we'll use Integral test for convergence

(Maclaurin–Cauchy test)

$lim_{n\to \infty}\sum \frac{1}{n} \thicksim \int_{1} ^\infty \frac{1}{x} =ln(x)|_1^\infty=\infty$