Quotient criteria and the harmonic series

[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?

 

[HallsofIvy]

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...

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

It does NOT "fulfill the quotient criteria".

So where else does it not work?

 

[foges]

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

 

[HallsofIvy]

What exactly do you think the ratio test says?

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

 

[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.

 

[Pjennings]

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