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If $K$ is a field complete with respect to a non-archimedean absolute value, then the $n$th term test (checking whether the $n$th term of a series goes to zero) is sufficient to check convergence of a series in $K$.

I was wondering if we have any results that guarantee, under additional hypothesis, whether a series in $\mathbb{R}$ converges by just making sure that the $n$th term goes to zero. An example would be the alternating series test. Are there others?

Arturo Magidin
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Derek Scavo
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    I suppose that Dirichlet's test is an example of what you're looking for: http://en.wikipedia.org/wiki/Dirichlet%27s_test – Jonas Meyer Dec 01 '10 at 04:21
  • For geometric series, $\sum ar^n$, the series converges if and only if $|r|\lt 1$, which is equivalent to $ar^n\to 0$ as $n\to\infty$. But you have to recognize its form (same with the Alternating Series Test). This is in contrast to the non-archimedean case, where the limit is dispositive by itself. – Arturo Magidin Dec 01 '10 at 05:24

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