Q. 6 - Calculus

Question

Let f(x) be a function that is continuous, positive, and decreasing on the interval $[1,\infty )$ such that $\underset{x\to \infty }{lim}f\left(x\right)=\alpha >0$. What can the divergence test tell us about the series $\sum _{k=1}^{\infty }f\left(k\right)$?

Step-by-Step Solution

Answer

By the divergence test, the series $\sum _{k=1}^{\infty }f\left(k\right)$ is continuous, positive and decreasing on the interval $[1,\infty )$ with limit $\underset{x\to \infty }{lim}f\left(x\right)=\alpha >0$ is divergent.

Step 1. Given Information.

The function:

$\underset{x\to \infty }{lim}f\left(x\right)=\alpha >0 on [1,\infty )$.

Step 2. Divergence test.

If the sequence $\left\{a_k\right\}$ does not converge to zero, then the series diverges.

$\underset{x\to \infty }{lim}f\left(x\right)=\alpha >0$

Step 3. By divergent test.

the value of the limit is greater than zero, so by the divergent test series $\sum _{k=1}^{\infty }f\left(k\right)$ is divergent.

So, by divergence test series, a function f(x) that is continuous, positive, and decreasing on the interval $[1,\infty )$ such that $\underset{x\to \infty }{lim}f\left(x\right)=\alpha >0$ is divergent.