Convergence and Divergence Question 4

Let $f$ be a positive, continuous, decreasing function such that $a_n=f(n)$ . If $\displaystyle \sum_{n=1}^\infty a_n$ converges to $k$ , which of the following must be true?

\lim\limits_{n\to\infty} a_n=k

\displaystyle \int\limits_1^n f(x)   dx = k

\displaystyle \int\limits_1^\infty f(x)   dx

converges.

\displaystyle \int\limits_1^\infty f(x)   dx = k

Approach

The integral test states that for a positive, continuous, and decreasing function,

\sum_{n=1}^\infty a_n \hskip{2em} \text{ and } \hskip{2em}\int\limits_1^\infty f(x)   dx

either both converge or diverge.

The test does not tell us the value the integral or series converges to.