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 February 23rd, 2018, 04:40 AM #1 Banned Camp   Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 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.
 February 23rd, 2018, 08:21 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 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.
 February 23rd, 2018, 08:29 AM #3 Senior Member   Joined: Nov 2011 Posts: 224 Thanks: 2 Nice question brother... Last edited by shaharhada; February 23rd, 2018 at 08:50 AM.
 February 23rd, 2018, 08:48 AM #4 Banned Camp   Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 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.
 February 23rd, 2018, 09:26 AM #5 Global Moderator   Joined: Dec 2006 Posts: 19,527 Thanks: 1750 If you're dealing with sequences of numbers, the subjects are mathematically equivalent.
 February 23rd, 2018, 10:44 AM #6 Banned Camp   Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 How I prove this: 1/1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... = ln(2). Thanks from Country Boy
 February 23rd, 2018, 11:03 AM #7 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,122 Thanks: 914 Is that a problem given out by your math teacher?
 February 23rd, 2018, 11:17 AM #8 Banned Camp   Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 No. Is it not solvable? Is this question is very hard ?! it is mark by two stars!!!
 February 23rd, 2018, 12:27 PM #9 Global Moderator   Joined: Dec 2006 Posts: 19,527 Thanks: 1750 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.
 February 24th, 2018, 02:29 AM #10 Banned Camp   Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 Thanks!!! Thanks from greg1313

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