12.3 Infinite Sequences and Series

Infinite sequence – a sequence that has infinitely many terms. Limits can be used to determine if a sequence approaches a value.

When any positive power of n appears only in the denominator of a fraction and n approaches infinity, the limit equals zero.

Ex 1 Estimate the limit of

For sequences with more complicated general forms For sequences with more complicated general forms. Applications of the following limit theorems can make the limit easier to find.

Ex 2 Find the limit

Ex 3 find the limit

Limits don’t exist for all infinite sequences Limits don’t exist for all infinite sequences. If the absolute value of a sequence becomes arbitrarily great or if the terms don’t approach a value the sequence has no limit. EX 4

Ex 5

When n is even, (-1)n = 1 and when n is odd, (-1)n = -1 When n is even, (-1)n = 1 and when n is odd, (-1)n = -1. Therefore the sequence would have no limit.

Sum of an Infinite Series – if Sn is the sum of the first n terms and S is a number such that S – Sn approaches zero as n increases without bound, then the sum of the infinite series is S.

Sum of an Infinite Geometric Series

Ex 6 find the sum of 60 + 24 + 9.9 + …

Ex 7 write 0.762762762…as a fraction