So I have the problem

$$\sum_{n = 1}^\infty (-1)^n \frac{n}{n+5}$$ and I have to figure out if the series converges or diverges. I start out by testing the conditions of the Alternating Serie Test, only to find that the limit of n as $|a_{n}|$ approaches infinity is 1, and as a result we cannot use the Alternating Series Test. So my book says to just use the test for divergence, and it has this as an answer:

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However, nowhere in my book or notes (unless I missed it) does it say that if $lim_{n\to \infty}|a_{n}|$ is not equal to zero, then the series is divergent. It only says that if $lim_{n\to \infty}a_{n}$ is not equal to zero, then the series is divergent, without the absolute value symbols. So I'm basically wondering how they made the test for divergence work here.

  • $\begingroup$ If $\lim_{n\to\infty} |a_n|\neq0$, can $\lim_{n\to\infty}a_n=0$? The answer is "No." and the reason is simple: $|0|=0$. $\endgroup$ – Clayton Oct 17 '13 at 2:44
  • $\begingroup$ You have an alternating series of -1 and 1 using division of highest power, hence divergence. The alternating series is what makes this divergent. $\endgroup$ – Display Name Oct 17 '13 at 2:44

If $\lim\limits_{n\to\infty}|a_n| = L \neq 0$, then either

  • $\lim\limits_{n\to\infty}a_n = L$,
  • $\lim\limits_{n\to\infty}a_n = -L$, or
  • $\lim\limits_{n\to\infty}a_n$ does not exist.

In each of these three cases, $\lim\limits_{n\to\infty}a_n \neq 0$, so the series diverges.

Note that $\lim\limits_{n\to\infty}|a_n| = 0$ if and only if $\lim\limits_{n\to\infty}a_n = 0$.

Therefore the divergence test could be rephrased as: If $\lim\limits_{n\to\infty}|a_n| \neq 0$, then $\sum\limits_{n=1}^{\infty}a_n$ diverges.

  • $\begingroup$ Ah, that makes perfect sense. That's why it is a condition of the Alternating Series Test that $\lim\limits_{n\to\infty}|a_n| = 0 $? So for an alternating series test, just seeing that the limit of $|a_n|$ is not zero is enough to let you know it diverges? $\endgroup$ – FrostyStraw Oct 17 '13 at 2:53
  • $\begingroup$ Knowing that the limit of $|a_n|$ is non-zero is enough to say the series diverges, whether it is an alternating series or not. The reason why we phrase the Alternating Series Test in terms of $|a_n|$ is that the terms alternate between positive and negative. $\endgroup$ – Michael Albanese Oct 17 '13 at 2:55
  • $\begingroup$ Yes, because otherwise it won't be getting closer and closer to 0. It will be taking and adding a greater number each time which will cause it to oscillate past 0 each time by a greater amount. $\endgroup$ – Display Name Oct 17 '13 at 2:56

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