Quotient criteria and the harmonic series

**foges** · Jul3-09, 06:16 AM

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

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

applied to the harmonic series:

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

So where else does it not work?

**HallsofIvy** · Jul3-09, 06:36 AM

Originally Posted by foges

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

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

applied to the harmonic series:

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

??? No!
$LaTeX Code: \\lim_{k\\to\\infty}|\\frac{k}{k+1}|= 1$ !

It does NOT "fulfill the quotient criteria".

So where else does it not work?

**foges** · Jul3-09, 07:31 AM

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

**HallsofIvy** · Jul3-09, 08:23 AM

What exactly do you think the ratio test says?

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

**Gib Z** · Jul6-09, 08:10 AM

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.

**Pjennings** · Jul6-09, 10:53 AM

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