[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.16,0:00:02.79,Default,,0000,0000,0000,,In the previous example,
Dialogue: 0,0:00:02.79,0:00:08.04,Default,,0000,0000,0000,,we talked about two different sequences \Nthat occur inside an infinite series.
Dialogue: 0,0:00:08.04,0:00:10.57,Default,,0000,0000,0000,,There's the sequence of individual terms
Dialogue: 0,0:00:10.57,0:00:13.50,Default,,0000,0000,0000,,(those are the pieces that \Nare being added together);
Dialogue: 0,0:00:13.50,0:00:16.99,Default,,0000,0000,0000,,and then there's also the \Nsequence of partial sums,
Dialogue: 0,0:00:16.99,0:00:21.55,Default,,0000,0000,0000,,where S-n means \Nthe sum of the first n terms.
Dialogue: 0,0:00:21.55,0:00:25.85,Default,,0000,0000,0000,,So S-4 would be the sum of\Nthe first four terms and so on.
Dialogue: 0,0:00:25.85,0:00:31.85,Default,,0000,0000,0000,,The idea is, if we look at the sequence \Nof partial sums or the running total,
Dialogue: 0,0:00:31.85,0:00:39.47,Default,,0000,0000,0000,,we can say, if the limit of that \Nsequence (as n goes to infinity)
Dialogue: 0,0:00:39.47,0:00:50.44,Default,,0000,0000,0000,,is equal to some number, \Nwhich we call S, then the series
Dialogue: 0,0:00:50.44,0:00:56.90,Default,,0000,0000,0000,,sum as n goes from \None to infinity of ‘a’-n,
Dialogue: 0,0:00:56.90,0:01:01.18,Default,,0000,0000,0000,,we say the series converges.
Dialogue: 0,0:01:01.18,0:01:07.27,Default,,0000,0000,0000,,And we actually say that the value \Nof that sum is this value right here.
Dialogue: 0,0:01:09.54,0:01:12.97,Default,,0000,0000,0000,,We say that the series converges to S.
Dialogue: 0,0:01:12.97,0:01:15.63,Default,,0000,0000,0000,,If the limit does not exist—
Dialogue: 0,0:01:15.63,0:01:24.51,Default,,0000,0000,0000,,So if limit as n goes to infinity \Nof S-n does not exist,
Dialogue: 0,0:01:24.51,0:01:38.09,Default,,0000,0000,0000,,we say the series converges--\N[corrects self] Sorry, diverges.\N