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.