Convergence of \sum \frac{r^k}{k^r} for Positive Numbers r

[zeion]

Homework Statement

Find the positive numbers r for which $$\sum \frac{r^k}{k^r}$$ converges.

Homework Equations

The Attempt at a Solution

What method do I use for this?

$$\frac {a_{k+1}}{a_k} = \frac {r^{k+1}}{(k+1)^r} \cdot \frac{k^r}{r^k}<br /> = r \cdot (\frac{1}{k})^{\frac{r}{k}}$$

 

[Mark44]

Homework Statement

Find the positive numbers r for which $$\sum \frac{r^k}{k^r}$$ converges.

Homework Equations

The Attempt at a Solution

What method do I use for this?

$$\frac {a_{k+1}}{a_k} = \frac {r^{k+1}}{(k+1)^r} \cdot \frac{k^r}{r^k}<br /> = r \cdot (\frac{1}{k})^{\frac{r}{k}}$$

The second and third expressions aren't equal...

 

[zeion]

Ok so I have $$\frac {a_{k+1}}{a_k} = \frac {r^{k+1}}{(k+1)^r} \cdot \frac{k^r}{r^k}<br /> = r \cdot \frac{k^r}{(k+1)^r } = r \cdot (\frac{k}{(k+1)})^r$$ ?

 

[zeion]

then $$r \cdot (1 - \frac{1}{k+1})^r \to r \cdot 1^r$$ ?

 

[zeion]

then r needs to be < 1?

 

[Mark44]

What if r = -200? That's less than 1.

 

[zeion]

I thought if the ratio is < 1 then it converges?

 

[Mark44]

The ratio test is usually given in terms of absolute values. IOW,
$$\lim_n \frac{|a_{n + 1}|}{|a_n|}$$

For this problem it's given that r is a positive number, so since the ratio of those terms is less than 1, the series converges.

 

[zeion]

So how do I know the ratio is less than 1?

 

[Mark44]

I thought you already knew that. What I thought you did was to evaluate this limit:
$$\lim_{k \to \infty} \frac {a_{k+1}}{a_k} = \lim_{k \to \infty} \frac {r^{k+1}}{(k+1)^r} \cdot \frac{k^r}{r^k}= \lim_{k \to \infty} r \cdot \frac{k^r}{(k+1)^r } = \lim_{k \to \infty} r \cdot (\frac{k}{(k+1)})^r$$
What do you get for this limit? And what does that value imply for the test you are using?

 

[zeion]

I get $$r \cdot 1^r = r$$?
And r is positive

 

[Mark44]

Find the positive numbers r for which $$\sum \frac{r^k}{k^r}$$
converges.

From your work, what are the values of r that cause this series to converge?

 

[zeion]

I need the ratio to approach something < 1?
So since r is positive I need it to be 0 < r < 1?

 

[Mark44]

C'mon, show some confidence. Tell me, don't ask me.

 

[Mark44]

I need the ratio to approach something < 1?
So since r is positive I need it to be 0 < r < 1?

C'mon, show a little confidence in your ability. Tell me, don't ask me.