Answer to Question #103937 in Calculus for BIVEK SAH

Question #103937
Evaluate

3n
Lim ∑ rootn/(n+3r)^3/2
n→∞ r=0
1
Expert's answer
2020-03-12T13:17:02-0400

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

limnn(n)3/2=limn1nlim_{n\to \infty}\sum \frac{\sqrt{n}}{(n)^{3/2}}=lim_{n\to \infty}\sum \frac{1}{n}

1/n>01/n>0 and decreasing with n, then we'll use Integral test for convergence

(Maclaurin–Cauchy test)

limn1n11x=ln(x)1=lim_{n\to \infty}\sum \frac{1}{n} \thicksim \int_{1} ^\infty \frac{1}{x} =ln(x)|_1^\infty=\infty


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