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Limit of Geometry Sequence and Series Motivation:
For geometry sequence A, Ar, Ar2, Ar3, Ar4, …, Arn, … Does this sequence has limit? Or under which condition that the limit of this sequence exists? Furthermore, the geometry series
Sn = A + Ar + Ar2 + Ar3 + … + Arn
What will happen if ? It involves with the sum of infinitely many terms.
Notation ( Summation ) We introduce a notation here for the sum of a sequence up to n terms:
= The notation means when i runs from 1 to n, you just add all of those terms for every possible i. We use some examples to explain.
Example 1: = n When i changes, 1 is still equal to 1. And after i runs from 1 to n, we have n 1s. So, the sum of those 1s is n.
Example 2: = 1 + 2 + 3 + 4 + … + n = n(n+1)/2 In this example, when i=1, we have a term 1; when i=2, we have 2; … Thus, the result is the sum from 1 to n.
Theorem: when |r| < 1 . Proof: For any >0, if we have | rn | < log | rn | < log n log |r| < log n > ( because log|r| < 0 when |r| < 1 )
In other words, for any > 0 , we can choose an integer K such that K is the least integer that is larger than , and |rn| < when n > K .
Thus, .
Example: Sol: Cn = = =0 and =0
Thus, Cn = 5
Theorem ( Convergence of Geometry Series ) Let Sn = . The limit of {Sn} exists only when | r | < 1 . Furthermore, if 0< |r| < 1, then
=
Proof: Case I: r 1, then Sn = =
{Sn} diverges when |r| > 1. When |r| < 1, . Thus, =
Case II: r=1, then Sn = n . In this case , {Sn} diverges .
Example: 0.1 + 0.01+ 0.001+ 0.0001 + …. ( infinitely many terms ) Sol: Let Sn = Then = = And this is the fact we know since long time ago that = 0.1111111…
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