# Convergence Test

1. Dec 3, 2007

### rocomath

$$\sum_{n=1}^{\infty}\frac{(-1)^{n}n}{n^{2}+25}$$

Ratio Test

$$\lim_{n\rightarrow\infty}|\frac{(-1)^{n+1}(n+1)(n^{2}+25)}{[(n+1)^{2}+25](-1)^{n}n}|$$

$$\lim_{n\rightarrow\infty}|\frac{n^{3}+n^{2}+25n+25}{ n^{3}+2n^{2}+26n}|=1$$

Thus, the Ratio Test is inconclusive. So what should my next step be, or other suggestions? Hmm ...

Thanks!

2. Dec 3, 2007

### Dick

How about an alternating series check?

3. Dec 4, 2007

### HallsofIvy

Staff Emeritus
In fact, since that is NOT a series of positive numbers, the ratio test doesn't apply anyway!

4. Dec 4, 2007

### rs1n

Just because contains non-positive terms does not mean one cannot apply the ratio test, since it compares the ratio of the absolute value of $$a_{n+1}$$ and $$a_n$$. His first attempt is fine; he just happened to have a series for which the ratio test is inconclusive.

5. Dec 4, 2007

### sutupidmath

Then can u show us a proof that shows that the ratio test is consistent and applies even when a series contains non-positive terms??

6. Dec 4, 2007

### morphism

I suppose you believe the ratio test holds for series whose terms are nonnegative. Suppose then we apply it to the series $\sum |a_n|$: If $\lim |a_{n+1}|/|a_n| < 1$, then $\sum |a_n|$ converges. But this in turn implies that $\sum a_n$ converges. This follows from the completeness of the real numbers, i.e. that every cauchy sequence of reals convereges.

To see this, let $S_n = a_1 + a_2 + ... + a_n$. Then for $n \geq m$,
$$|S_n - S_m| = |a_{m+1} + ... + a_n| \leq |a_{m+1}| + ... + |a_n|$$

If $\sum |a_n|$ converges, we can make the term on the right as small as we want. So $(S_n)_n$ is cauchy, and consequently $\sum a_n = \lim S_n < \infty$.

Last edited: Dec 4, 2007
7. Dec 4, 2007

### HallsofIvy

Staff Emeritus
I agree that what I said at first was misleading, possibly just completely wrong!

Of course, if the ratio test, applied to |an| showed that it converged, that would show that the series is absolutely convergent which immediately implies that the series is convergent.

If the ratio test does not work, if the limit of the ratio is 1 or even greater than 1, it is still possible that the original series converges. As Dick said originally, it is far better to apply the "alternating series test" here. If |an| is decreasing, then the series converges.

Last edited: Dec 4, 2007
8. Dec 6, 2007

### sutupidmath

OOh yeah, i forgot all about that. I just did not reflect on this at all. I've got to whatch my mouth next time.