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I have a math exam tomorrow and I have done all homeworks and have reviewed heavily, but I have to admit that I am still confused.

What I know: a sequence is the complete set of terms, for example {1,1/2,1/4,1/8,1/16}, whereas a sequence is a summation of the partial sums. In this case, you would add up the partial sums. However, can a sequence converge to a non-zero number?

I understand the different tests for calculating convergence, and can do improper integrals and the like, I just want a very in-depth understanding. Thanks much.

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    It seems to me that you are confusing series and sequences. Given a series $\sum a_n$, you get two sequences: the sequence the terms, $a_1,a_2,a_3,\ldots$, and the sequence of partial sums, $a_1, a_1+a_2, a_1+a_2+a_3,\ldots$. Convergence of the *series* is convergence of the sequence of partial sums. By the Divergence Test, the if the series converges then the sequence of *terms* must converge to zero. So if the *terms* don't converge to zero (either they diverge or they converge to something else), then the *series* diverges.2011-11-07
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    It's the sequence which either converges or diverges, not the \*limit\* of a sequence. The limit is what the sequence converges to.2011-11-07
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    Thanks both, I appreciate it2011-11-07

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