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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. –  Arturo Magidin Nov 7 '11 at 2:01
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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. –  Srivatsan Nov 7 '11 at 2:02
    
Thanks both, I appreciate it –  Arthur Collé Nov 7 '11 at 12:18
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1 Answer

Answered in a comment by Arturo Magidin:

It seems to me that you are confusing series and sequences. Given a series $\sum a_n$, you get two sequences:

  • the sequence of the terms, $a_1,a_2,a_3,\dots$
  • the sequence of partial sums, $a_1,a_1+a_2,a_1+a_2+a_3,\dots$

Convergence of the series is convergence of the sequence of partial sums. By the Divergence Test, 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.

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