# Infinite Series

## Homework Statement

Is the series from n=1 to infinity of 3/n converging or diverging?

## The Attempt at a Solution

Since 3/n is not a geometric series, my guess is that we can just use the Test for Divergence and take it's limit to see if it's converging or diverging. As n->infinity, 3/n -> 0 and lim = 0, so it's converging.

However, I am not sure if this is right way to go about it.

Tom Mattson
Staff Emeritus
Gold Member
Since 3/n is not a geometric series,

Correct.

my guess is that we can just use the Test for Divergence and take it's limit to see if it's converging or diverging.

Not a bad guess, but beware that the Test for Divergence cannot tell you if a series converges (hence, its name).

As n->infinity, 3/n -> 0 and lim = 0, so it's converging.

Wrong. The Test for Divergence says that:

$$\lim_{n\rightarrow\infty}a_n \neq 0 \Rightarrow \sum_{n=1}^\infty a_n$$ diverges.

Equivalently, it says that:

$$\sum_{n=1}^\infty a_n$$ converges $$\Rightarrow \lim_{n\rightarrow\infty}a_n = 0$$

If the limit is zero, then the test yields no information and you have to use another test.

So I would have to treat it as a Harmonic Series?

Tom Mattson
Staff Emeritus