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Can a series have a limit and be divergent? I'm confused about the difference between divergence and a series not having a limit. A ck-12 calculus book stated they are different concepts.

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The limit could be $\infty$ for example, in which case we still say it's divergent. –  Najib Idrissi Jun 18 '13 at 19:04
    
If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $\sum_{n = 1}^\infty a_n$ converges if and only if the sequence of partial sums $s_k = \sum_{n = 1}^k a_n$ has a limit as $k$ approaches $\infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n \to \infty$. –  Ben Passer Jun 18 '13 at 21:09

1 Answer 1

If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.

The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $\sum_{n \ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,\ldots$.

The partial sums obtain a finite limit if and only if the series is convergent.

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