Alternating Series Test basically says if the limit of the nth term is 0, and the series is decreasing (redundant?), then the series converges.

The Ratio test says that if limit of $|\frac{a_n+1}{a_n}|$ < 1 then it converges absolutely.

First, the fraction part is simply another way to say the series is decreasing (The next number is less than the previous one)

So, what's the difference?

Also, what is the point of converges absolutely? Does that just mean if you are ignoring the +/- aspect, are are only thinking about the convergence of the numbers in the series?

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    $\begingroup$ The sequence $a_n=\frac 1n$ is decreasing, yet the sum diverges (and the limit computed by the ratio test is $1$). $\endgroup$
    – lulu
    Apr 1, 2017 at 17:53
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    $\begingroup$ If a series converges absolutely, we have by definition $\sum |a_n|$ converges, and we may safely rearrange the terms of $\sum a_n$. $\endgroup$ Apr 1, 2017 at 17:53
  • $\begingroup$ Take a look at the Riemann rearrangement theorem. $\endgroup$
    – Masacroso
    Apr 1, 2017 at 18:01
  • $\begingroup$ @Jack. Please not that if an alternating sequence converges to zero, that does not imply convergence for the series. It is needed that (ultimately) every next term is (in absolute terms) smaller than the previous. $\endgroup$
    – imranfat
    Apr 1, 2017 at 18:11

1 Answer 1

  1. The decreasing to zero is not redundant, you can have a sequence that decreases but not to $0$, say $a_n=1+\frac{1}{n}$. Note that $\sum a_n$ diverges.

  2. The ratio test cares about the rate at which the sequence decreases, just being decreasing is not enough (for example, $\sum\frac{1}{n}$ does not converge). It also does not take into account any signs, as you can see from the absolute value. The common example is that the alternating series test tells you that the series $\sum\frac{(-1)^n}{n}$ converges, the ratio test tells you nothing in this case.

  3. Absolute convergence is very important. Conditional convergence is quite delicate, requiring cancellations to occur. Its actually the case that any conditionally convergent series can be rearranged to give any value (Riemann series theorem ). In particular, it really matters the order in which you sum the terms, which is not the case for absolutely convergent series.


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