Integral Test — Question 7

*Question 7 Consider $$\sum_{n=1}^{\infty}\frac1{n^2+4}.$$

1. Verify positivity, continuity, and decreasing behavior for $f(x)=1/(x^2+4)$.

2. Use the Integral Test to classify the series, showing the arctangent antiderivative and improper limit.

3. Write explicit upper and lower bounds for the remainder after $N$ terms.

Your remainder bounds should contain $N$ and inverse tangent functions.

*Question 7 – Solution

Define $f(x)=1/(x^2+4)$ for $x\ge1$. It is continuous and positive, and $$f'(x)=-\frac{2x}{(x^2+4)^2}<0,$$ so it decreases.

$$\int_1^{\infty}\frac{dx}{x^2+4} =\frac12\left[\arctan\left(\frac x2\right)\right]_1^{\infty} =\frac12\left(\frac\pi2-\arctan\frac12\right)<\infty.$$ Therefore the series converges.

For $A>0$, $$\int_A^{\infty}\frac{dx}{x^2+4} =\frac12\left(\frac\pi2-\arctan\frac A2\right).$$ Consequently, $$\frac12\left(\frac\pi2-\arctan\frac{N+1}{2}\right) \le R_N\le \frac12\left(\frac\pi2-\arctan\frac N2\right).$$