Showing a sequence converges if sequences of terms satisfy a condition

Question

Suppose we have a series $\sum a_n$ so that $(a_n)$ is a decreasing sequence. Can we conclude that $\sum a_n $ is convergent??

Answer

I think we can use ratio test. Since $a_n \geq a_{n+1}$ for all $n$, then

$$ \frac{ a_{n+1}}{a_n} \leq 1 $$

so the series converges by ratio test. However, Im not quite sure. I think we must assume that $a_n$ is never zero, say $a_n > 0$ for all $n$. Then result would follow. Is this correct?

Answer 1

You must have $a_n>0$, else $a_n=-n$ would be a counterexample.

But that's not enough, $a_n=1+\frac{1}{n}$ is decreasing, but the sum is obviously not convergent.

If you have $a_n>0$ and $\lim_{n\to\infty} a_n=0$ you get closer, but it's still not enough, the harmonic series being a counterexample.

Answer 2

We cannot conclude that.

Let $a_n = 1+(1/2)^n$ which is decreasing.

$$ \int_1^\infty 1+(1/2)^x dx <\sum_{n=0}^\infty a_n$$

But $\int_1^\infty 1+(1/2)^x dx = [x - 2^{-x}/log(2) ]_1^{\infty}$ which is divergent.