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SEQUENCES AND SERIES Arithmetic

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Definition A series is an indicated sum of the terms of a sequence. Finite Sequence: 2, 6, 10, 14 Finite Series:2 + 6 + 10 + 14

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Formula The sum of the first n terms of an arithmetic series is:

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Example 1.Find S 25 of the arithmetic series 11 + 14 + 17 + 20+ …

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Example First you have to find the 25 th term. Now you can find the sum of the first 25 terms.

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Examples Continued 2.Find the sum of the arithmetic series 5+9+13+…+153.

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Examples Continued 2.Find the sum of the arithmetic series 5+9+13+…+153. First you must determine how many terms you are adding. Now you can find the sum of the first 38 terms.

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Series The sum of the first n terms of a geometric series is:

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Examples 1.Find the sum of the first 10 terms of the geometric series 2 – 6 + 18 – 54 +…

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Examples

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INFINITE SERIES Geometric

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Converge Vs. Diverge An infinite series converges if the ratio lies between -1 and 1. Do the following series converge or diverge? 1.3+9+27+… 2.16+4+1+1/4+…

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Formula The sum, S, of an infinite geometric series where -1<r<1 is given by the following formula:

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Examples Find S 1, S 2, S 3, S 4, and the infinite sum if it exists. 1. 8+4+2+1+…2. 1+3+9+27+…

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Examples Find S 1, S 2, S 3, S 4, and the infinite sum if it exists. 1. 8+4+2+1+…2. 1+3+9+27+… Series Diverges

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SIGMA NOTATION

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Sigma Notation Consider the series: 6 + 12 + 18 + 24 + 30 or another way, 6(1) + 6(2) + 6(3) + 6(4) + 6(5) = 6n

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Sigma Notation In Sigma Notation, that series would look like: It is read as “the sum of 6n for values of n from 1 to 5”

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PARTS OF SIGMA Summand – 6n Index – n Lower Limit – 1 Upper Limit – 5

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Sigma Notation Continued The lower limit is the first number that is being substituted in for n. The upper limit is the last number that is being substituted in for n.

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EXAMPLES Write in expanded form and find the sum: 1. 2.

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EXAMPLES Write in expanded form and find the sum: 1. 2. Lower Limit Upper Limit Lower Limit

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Express using Sigma Notation: 3. 10 + 15 + 20 + … + 100 4. 5 + 10 +20 + 40 + 80 + 160

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Express using Sigma Notation: 3. 10 + 15 + 20 + … + 100 4. 5 + 10 +20 + 40 + 80 + 160

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Express using Sigma Notation: 3. 10 + 15 + 20 + … + 100 4. 5 + 10 +20 + 40 + 80 + 160

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5. 1 + 4 + 9 + 16 + 25 + … 6. 1 + ½ + 1/3 + ¼ + …

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5. 1 + 4 + 9 + 16 + 25 + … 6. 1 + ½ + 1/3 + ¼ + …

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5. 1 + 4 + 9 + 16 + 25 + … 6. 1 + ½ + 1/3 + ¼ + …

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