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12.3 Infinite Sequences and Series

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

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

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Ex 1 Estimate the limit of

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

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Ex 2 Find the limit

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Ex 3 find the limit

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Limits dont exist for all infinite sequences. If the absolute value of a sequence becomes arbitrarily great or if the terms dont approach a value the sequence has no limit. EX 4 EX 4

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Ex 5

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

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

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Sum of an Infinite Geometric Series

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Ex 6 find the sum of …

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Ex 7 write …as a fraction

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