Integral Test — Question 9

*Question 9 Consider the shifted $p$-series $$\sum_{n=1}^{\infty}\frac1{(n+3)^{5/4}}.$$

1. Verify the Integral Test hypotheses for $f(x)=(x+3)^{-5/4}$.

2. Evaluate the associated improper integral and classify the series.

3. Obtain two-sided bounds for $R_N$.

4. Explain why a fixed horizontal shift does not alter the underlying $p$-series verdict.

*Question 9 – Solution

Set $f(x)=(x+3)^{-5/4}$ for $x\ge1$. It is continuous and positive, and $$f'(x)=-\frac54(x+3)^{-9/4}<0,$$ so it decreases.

$$\int_1^{\infty}(x+3)^{-5/4}\,dx =\left[-4(x+3)^{-1/4}\right]_1^{\infty}=\frac4{\sqrt2}<\infty.$$ Thus the shifted $p$-series converges. The horizontal shift changes initial values but not the tail exponent $p=5/4>1$.

$$\boxed{\frac4{(N+4)^{1/4}}\le R_N\le\frac4{(N+3)^{1/4}}}.$$ Although convergent, the exponent is close to $1$, so the error decreases slowly.