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February 23rd, 2018, 04:40 AM   #1
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convergent sequence

A series is convergent if the sequence of its partial sums $\displaystyle \left(S_{1},\ S_{2},\ S_{3},\dots \right)$ tends to a limit.
How I know when the sequence of convergent sequence has a limit?
[If you give me a link that explains it or write some words, I'll be happy.]

Last edited by skipjack; February 23rd, 2018 at 09:14 AM.
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February 23rd, 2018, 08:21 AM   #2
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Normally one studies "convergence of sequences" before studying "convergence of series", so you should already know that!

But there are many different ways to determine the convergence of a sequence, so what you should do may vary with the specific series. Can you give an example of a series that you are concerned with?

Last edited by skipjack; February 23rd, 2018 at 09:15 AM.
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February 23rd, 2018, 08:29 AM   #3
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Nice question brother...

Last edited by shaharhada; February 23rd, 2018 at 08:50 AM.
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February 23rd, 2018, 08:48 AM   #4
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Why is the order important?
What's the different between the two subjects?
Is the "convergence of a sequence" the basis of "convergence of series"?

Last edited by skipjack; February 23rd, 2018 at 09:19 AM.
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February 23rd, 2018, 09:26 AM   #5
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If you're dealing with sequences of numbers, the subjects are mathematically equivalent.
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February 23rd, 2018, 10:44 AM   #6
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How I prove this:
1/1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... = ln(2).
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February 23rd, 2018, 11:03 AM   #7
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Is that a problem given out by your math teacher?
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February 23rd, 2018, 11:17 AM   #8
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No. Is it not solvable?
Is this question is very hard ?!
it is mark by two stars!!!
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February 23rd, 2018, 12:27 PM   #9
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It's well-known that ln(1 + x) = x - x²/2 + x³/3 - . . . for -1 < x $\small\leqslant$1.

If you can use that, substitute x = 1.
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February 24th, 2018, 02:29 AM   #10
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Thanks!!!
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