February 23rd, 2018, 04:40 AM  #1 
Member Joined: Dec 2017 From: Tel Aviv Posts: 44 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: 2,966 Thanks: 807 
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: 184 Thanks: 2 
Nice question brother...
Last edited by shaharhada; February 23rd, 2018 at 08:50 AM. 
February 23rd, 2018, 08:48 AM  #4 
Member Joined: Dec 2017 From: Tel Aviv Posts: 44 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: 18,702 Thanks: 1527 
If you're dealing with sequences of numbers, the subjects are mathematically equivalent.

February 23rd, 2018, 10:44 AM  #6 
Member Joined: Dec 2017 From: Tel Aviv Posts: 44 Thanks: 3 
How I prove this: 1/1  1/2 + 1/3  1/4 + 1/5  1/6 + ... = ln(2). 
February 23rd, 2018, 11:03 AM  #7 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 11,847 Thanks: 766 
Is that a problem given out by your math teacher?

February 23rd, 2018, 11:17 AM  #8 
Member Joined: Dec 2017 From: Tel Aviv Posts: 44 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: 18,702 Thanks: 1527 
It's wellknown 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 
Member Joined: Dec 2017 From: Tel Aviv Posts: 44 Thanks: 3 
Thanks!!!


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