# Concerning alternating series: test for divergence fails (typo in the book).

Here is a series:

$$- \frac 2 5 + \frac4 6 - \frac 6 7 + \frac8 8 - \frac{10} 9 +\dots$$

The series is convergent (it says so in the back of the book) but the test for divergence fails:

We have, if $\lim_{n\to\infty}{a_n} \neq 0$, then the series is divergent. In this case we have $$a_n=(-1)^{n}\frac{2n}{n+4}.$$ The $\lim_{n\to\infty}{a_n} \neq 0$, so the series should be divergent. But I know it is convergent.

What am I doing wrong?

• If a series is convergent the general term goes to zero when $n$ goes to infinity, how can you conclude this? – DiegoMath Dec 9 '14 at 16:30
• @DiegoMath I have a theorem in my book – khajvah Dec 9 '14 at 16:32
• The theorem should be quoted in full. If it implies that the series in the OP converges, then the theorem is wrong, a serious problem. If the back of the book says the series converges, that is less serious, merely a typo. – André Nicolas Dec 9 '14 at 16:39
• @AndréNicolas Most probably the typo is the case. – khajvah Dec 9 '14 at 16:46
• The relevant theorem is that if $\lim_{n\to\infty}a_n\ne 0$, and also if the limit does not exist, then the series $\sum a_n$ diverges. – André Nicolas Dec 9 '14 at 16:51

Well, it does not converge: the term converges to $\pm2$, not to $0$, so at least by the Cauchy convergence criterion the sum does not converge.
• Re: "you transform the alternating sequence into one that does not alternate signs. Then, if the non-alternating sequence converges to zero": This part is not necessary. A sequence $a_n$ converges to zero and if only if the sequence $\left|a_n\right|$ converges to zero. (You may be conflating this with a different test, which states that any series $\sum_{n=1}^{\infty}a_n$ must necessarily converge if $\sum_{n=1}^\infty\left|a_n\right|$ does. That test, incidentally, is not specific to strictly alternating series.) – ruakh Dec 9 '14 at 21:36