Quotient criteria and the harmonic series

1. Jul 3, 2009

foges

Ok so you can't apply the quotient criteria to the harmonic series because:

$$lim_{k\to \infty}|\dfrac{a_{k+1}}{a_k}|$$

applied to the harmonic series:

$$lim_{k\to \infty}|\dfrac{1/(k+1)}{1/k}| = lim_{k\to \infty}|\dfrac{k}{k+1}| < 1$$
which does fullfill the quotient criteria, yet the harmonic series diverges...

So where else does it not work?

2. Jul 3, 2009

HallsofIvy

Staff Emeritus
??? No!
$$\lim_{k\to\infty}|\frac{k}{k+1}|= 1$$!

It does NOT "fulfill the quotient criteria".

3. Jul 3, 2009

foges

ok, so its the fact that it converges to 1 which makes it not work?

4. Jul 3, 2009

HallsofIvy

Staff Emeritus
What exactly do you think the ratio test says?

It looks to me like it does exactly what it claims to do!

5. Jul 6, 2009

Gib Z

The fact that the ratio test gives 1 is an inconclusive result - it does not tell us it converges or diverges. More is needed to show this series diverges.

6. Jul 6, 2009

Pjennings

If you are looking to establish the divergence of the harmonic series try using the integral test.