Integral Test — Question 1

*Question 1 Consider the series $$\sum_{n=1}^{\infty}\frac1{n^{3/2}}.$$

1. Define a continuous function $f$ with $f(n)=1/n^{3/2}$ and verify that it is positive, continuous, and decreasing on the required interval.

2. Use the Integral Test to determine whether the series converges or diverges.

3. If $s_N$ is the $N$th partial sum and $R_N=S-s_N$, derive both an upper and a lower bound for $R_N$.

Show the improper-integral limit and all antiderivative work.

*Question 1 – Solution

Let $f(x)=x^{-3/2}$ for $x\ge1$. It is continuous and positive, and $$f'(x)=-\frac32x^{-5/2}<0,$$ so it is decreasing. Also $f(n)=1/n^{3/2}$.

$$\int_1^{\infty}x^{-3/2}\,dx =\lim_{b\to\infty}\left[-2x^{-1/2}\right]_1^b=2.$$ The integral converges, so the series converges by the Integral Test.

For a positive decreasing $f$, $$\int_{N+1}^{\infty}f(x)\,dx\le R_N\le\int_N^{\infty}f(x)\,dx.$$ Therefore $$\boxed{\frac2{\sqrt{N+1}}\le R_N\le\frac2{\sqrt N}}.$$ The test proves convergence; these inequalities additionally measure the truncation error.