Q. 9 - Calculus

Question

Provide a more general statement of the integral test in which the function f is continuous and eventually positive, and decreasing. Explain why your statement is valid.

Step-by-Step Solution

Answer

The statement is valid because the tail of the series determines the convergence or divergence.

Step 1. Given Information.

The function f.

Step 2. The integral test.

If $f\left(x\right):[1,\infty )\to \mathrm{ℝ}$ is continuous, eventually positive and decreasing on $[1,\infty )$ , and $f_k$ is the sequence defined by $f_k=\left\{f\right(k\left)\right\}$ for every $k\in {\mathrm{\mathbb{Z}}}^{+}$ , then $\sum _{k=1}^{\infty }{f}_k and {\int }_1^{\infty }f\left(x\right)dx$ either both converge or diverge.

Step 3. To explain.

The statement is valid because the tail of the series determines the convergence or divergence.