An infinite arithmetic series can't ever converge at a finite value, because your terms can't ever approach 0 (without passing it toward positive/negative infinity, that is). However, under the right circumstances (|r| is less than 1), an infinite geometric series can meet that condition and therefore contain a finite sum value.
In the lecture portion, a proof takes the finite geometric series formula...
Sn = a1(1 - r^n)/(1 - r)
... and reworks it in such a way that, as n approaches infinity, r^n approaches zero (again, assuming that |r| is less than 1).
S = a1(1 - 0)/(1 - r)
S = a1(1)/(1 - r)
*S = a1/(1 - r)*
This is the formula we can use for an infinite geometric series, provided that |r| is less than 1.
Have I stated that |r| is less than 1 is necessary? I can't overstate it. If |r| was 1 or greater, than the series would diverge and have no finite sum, and this formula therefore does not apply.
This is because our terms need to shrink toward 0 as n approaches infinity, so any common ratio between -1 and 1 (exclusive) will be appropriate here.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
*PDF DOWNLOADS*
Textbook (7.4): https://smallpdf.com/file#s=39ac28d2-6c0d-4129-bf66-cd91bde3d279
Graph paper (scaled): https://docdro.id/flV4fYe
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
*TIMESTAMPS (separated by section)*
(0:00:00) Introduction
(0:00:49) Lecture overview
(0:11:19) Problem #1-2
(0:13:10) Problem #3-6
(0:31:04) Problem #7-14
(0:40:37) Problem #15-16
(0:44:51) Problem #17-18
(0:49:43) Problem #19-24
(0:57:36) Problem #25
(1:01:23) Problem #26
(1:02:29) Problem #27
(1:04:07) Problem #28
(1:19:09) Problem #29
(1:24:10) Problem #30
(1:25:39) Problem #31
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
*BIG IDEAS MATH (IM3) PLAYLIST*
https://www.youtube.com/playlist?list=PLsWOyuFufO779Te_I5J78GCU60qekMG-Z