# Thread: Proving the Ratio Test

1. ## Proving the Ratio Test

The question I've been given involves proving two of the three different scenarios of the Ratio Test. We've been given it as follows.

Let $\displaystyle \sum a_k$ be a series with positive terms and suppose that $\displaystyle \frac{a_{k+1}}{a_k}\rightarrow\lambda$.
1. If $\displaystyle \lambda<1$, then $\displaystyle \sum a_k$ converges. (this proof was given to us already)
2. If $\displaystyle \lambda>1$, then $\displaystyle \sum a_k$ diverges.
3. If $\displaystyle \lambda=1$, then the test is inconclusive.

The second and third parts need to be proven.

I believe I've gotten a good proof for the second one, where $\displaystyle \sum a_k$ diverges, though I could use a second opinion.

The third one, where the test is inconclusive, however, escapes me in terms of a proof. I could surely use a hand with that one.

2. 2. If $\displaystyle \lim_{n \to \infty} \, \frac{a_{n+1}}{a_n} = \lambda > 1$, then $\displaystyle a_n \le a_{n+1}$ for large n. Therefore, $\displaystyle \lim_{n \to \infty} \,a_n \not= 0 \Longrightarrow \sum a_n$ is divergent.

3. Take the p-series $\displaystyle \sum n^{-p}$

$\displaystyle \lim_{n \to \infty} \, \frac{a_{n+1}}{a_n}=1$ for any p.

3. Originally Posted by Runty
The question I've been given involves proving two of the three different scenarios of the Ratio Test. We've been given it as follows.

Let $\displaystyle \sum a_k$ be a series with positive terms and suppose that $\displaystyle \frac{a_{k+1}}{a_k}\rightarrow\lambda$.
1. If $\displaystyle \lambda<1$, then $\displaystyle \sum a_k$ converges. (this proof was given to us already)
2. If $\displaystyle \lambda>1$, then $\displaystyle \sum a_k$ diverges.
3. If $\displaystyle \lambda=1$, then the test is inconclusive.

The second and third parts need to be proven.

I believe I've gotten a good proof for the second one, where $\displaystyle \sum a_k$ diverges, though I could use a second opinion.

The third one, where the test is inconclusive, however, escapes me in terms of a proof. I could surely use a hand with that one.
For part three simply note that both series $\displaystyle \sum\limits_{n = 1}^\infty {\frac{1}{n}} \,\& \,\sum\limits_{n = 1}^\infty {\frac{1}{{n^2 }}}$ yield the same result from the ratio test.
But one diverges while the other converges.