Finding a Convergent Sequence with a Limit of 1

[DPMachine]

Homework Statement

Give an example of a sequence (a_n) so that lim_{n\rightarrow\infty} \left|a_{n+1}/a_{n}\right| =1 and \sum^{\infty}_{n=1} a_{n} converges

Homework Equations

(Maybe relevant, maybe not)
Theorem which states:

If \sum^{\infty}_{n=1} a_{n} converges, then lim_{n\rightarrow\infty} a_{n} =0

The Attempt at a Solution

I'm having trouble coming up with \sum^{\infty}_{n=1} a_{n} that converges...

Since lim_{n\rightarrow\infty} a_{n} =0 doesn't imply the convergence of \sum^{\infty}_{n=1} a_{n} (the theorem only works the other way around), I'm not sure where to start.

Any hint will be appreciated. Thank you.

 

[Landau]

I'm having trouble coming up with \sum^{\infty}_{n=1} a_{n} that converges...

You can certainly come up with some converging sum, ignoring the other requirement for a moment?

a_n=\frac{1}{n} satisfies \left|\frac{a_{n+1}}{a_n}\right|\to 1, but \sum^{\infty}_{n=1} a_{n} does not converge. Can you modify this example such that it does?

 

[DPMachine]

You can certainly come up with some converging sum, ignoring the other requirement for a moment?

a_n=\frac{1}{n} satisfies \left|\frac{a_{n+1}}{a_n}\right|\to 1, but \sum^{\infty}_{n=1} a_{n} does not converge. Can you modify this example such that it does?

Sorry, I don't think my question was clear. I'm having trouble understanding what makes certain partial sums converge/not converge.

So yeah, a_n=\frac{1}{n} does satisfy \left|\frac{a_{n+1}}{a_n}\right|\to 1, but why wouldn't \sum^{\infty}_{n=1} a_{n} converge? Isn't 1/n approaching zero?

 

[Bohrok]

Not all series converge whose sequence approaches 0. You can read about the sequence 1/n here
http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)

Look especially at the p-series on that page and see if that helps you find a sequence that fits the problem you posted.

 

[Mark44]

The purpose of this exercise seems to be exploring the edge case of the ratio test. For the terms in an infinite series \sum a_n, you look at the limit
\lim_{n \rightarrow \infty}\left|\frac{a_{n + 1}}{a_n}\right|~=~L
If L < 1, the series converges absolutely.
If L > 1, the series diverges.
If L = 1, or no limit exists, the test is inconclusive.

This problem seems to be about that third possibility, where L = 1.

 

[DPMachine]

The purpose of this exercise seems to be exploring the edge case of the ratio test. For the terms in an infinite series \sum a_n, you look at the limit
\lim_{n \rightarrow \infty}\left|\frac{a_{n + 1}}{a_n}\right|~=~L
If L < 1, the series converges absolutely.
If L > 1, the series diverges.
If L = 1, or no limit exists, the test is inconclusive.

This problem seems to be about that third possibility, where L = 1.

Okay, I think I understand it... so if L=1, it's possible for \sum a_{n} to either converge or diverge.

So, for example, when a_{n} is an alternating harmonic series (from the wiki article above):

then \sum a_{n} would converge and also have L=1.

On the other hand, if a_{n} is something like a_{1}=1, a_{n+1}=a_{n} so that (a_{n}) = (1, 1, 1, 1, 1, ...)

then \sum a_{n} does not converge (since it goes to infinity) but still have L=1.

Is that right?

 

[Mark44]

Right.