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Geometric Series

Geometric Series. section 12.3. DEFINITIONS. Geometric Series is indicated sum of the terms of a geometric sequence Ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. 5. ∑. 3 n. k = 1. 5. 3 + 6 + 9 + 12 + 15 = ∑ 3 k. k = 1.

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Geometric Series

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  1. Geometric Series section 12.3 12.2 - Geometric Series

  2. DEFINITIONS Geometric Seriesis indicated sum of the terms of a geometric sequence Ratiois a quantity that denotes the proportional amount or magnitude of one quantity relative to another 12.2 - Geometric Series

  3. 5 ∑ 3n k =1 5 3 + 6 + 9 + 12 + 15 = ∑3k k = 1 Equation of Series Is read as “the sum from nequals 1 to 5 of 3n.” upper limit of summation index of summation lower limit of summation 12.2 - Geometric Series

  4. Steps Identify the number of terms, lower and upper limit, and common ratioof the summation Then, identify the missing terms by plugging the given equation Apply the geometric summation notation equation 12.2 - Geometric Series

  5. Example 1 Find the indicated sum for the geometric series of s8 = 1 + 2 + 4 + 8… Step 1: Identify the number of terms, lower and upper limit of the summation 12.2 - Geometric Series

  6. Example 1 Find the indicated sum for the geometric series of s8 = 1 + 2 + 4 + 8… Step 2: Find the common ratio. 12.2 - Geometric Series

  7. Example 1 Find the indicated sum for the geometric series of s8 = 1 + 2 + 4 + 8 + 16 … Step 3: Find s8 with a1 = 1, r = 2, and n = 8 12.2 - Geometric Series

  8. Example 2 Find the indicated sum for the geometric series of s5 = 7 + 14 + 28 …. 12.2 - Geometric Series

  9. Your Turn Find the indicated sum for the geometric series of s9 = –3 + 6 + –12… 12.2 - Geometric Series

  10. Example 3 Evaluate 12.2 - Geometric Series

  11. Example 3 Evaluate 12.2 - Geometric Series

  12. Example 4 Evaluate 12.2 - Geometric Series

  13. Your Turn Evaluate 12.2 - Geometric Series

  14. Example 5 Evaluate 12.2 - Geometric Series

  15. Definitions Finite series has a limited number of terms, such as {1, 2, 3, 4}. Infinite series which continues without end such as {1, 2, 3, 4, …} 12.2 - Geometric Series

  16. Is it Finite vs Infinite? 1. 3 + 6 + 9 + 12 + 15 2. 3 + 6 + 9 + 12 + … 3. -1/3, 1/5, -1/9, 1/17, and -1/33. 4. 32, 16, 8, 4, 2, 1, … • Finite • Infinite • Finite • Infinite 12.2 - Geometric Series

  17. Definitions Infinite Geometric Serieshas unlimited terms Convergentseries is where the ratio is LESS than 1; it DOES have an infinite sum Divergentseries is where the ratio is MORE than 1; it DOESNOT have an infinite sum 12.2 - Geometric Series

  18. Convergent vs. Divergent Formula: Determine whether these series is eitherconvergent or divergent. Find the ratio to identify whether it is convergent or divergent. Ratio: 4/2 = 2 Ratio: 16/32 = 1/2 Divergent Convergent 12.2 - Geometric Series

  19. Examples To determine whether the series is convergent or divergent, find the ratio. 1. 10 + 1 + 0.1 + 0.01 + … 2. 4 + 12 + 36 + 108 + … 3. Convergent, r = 1/10 Divergent, r = 3 Divergent, r = 3/2 12.2 - Geometric Series

  20. Sum of infinite series formula Formula: What’s so different about this equation than the other geometric equation? Rules of r: If |r| < 1, use the formula (converges) If |r| > 1, it does not have a sum (diverges) 12.2 - Geometric Series

  21. Is it infinite or not? Given: 12.2 - Geometric Series

  22. Steps • Identify whether the INFINITE sequence geometric or not • Determine the ratio • Apply the equation, 12.2 - Geometric Series

  23. Example 7 Given:Find the sum of the infinite geometric series, if it exists. Step 1: Identify whether the INFINITE sequence geometric or not infinite geometric series 12.2 - Geometric Series

  24. Example 7 Given:Find the sum of the infinite geometric series, if it exists. Step 2: Determine the ratio 12.2 - Geometric Series

  25. Example 7 Given:Find the sum of the infinite geometric series, if it exists. Step 3: Apply the equation 12.2 - Geometric Series

  26. Example 8 Given:Find the sum of the infinite geometric series, if it exists. 12.2 - Geometric Series

  27. Example 9 Given: Find the sum of the infinite geometric series, if it exists. 12.2 - Geometric Series

  28. Your Turn Given: Find the sum of the infinite geometric series, if it exists. 12.2 - Geometric Series

  29. Example 10 Given: Find the sum of the infinite geometric series, if it exists. 12.2 - Geometric Series

  30. Example 11 Given: Find the sum of the infinite geometric series, if it exists. 12.2 - Geometric Series

  31. Example 12 Write 0.23232323… as a fraction in simplest form Step 1: Identify whether the INFINITE sequence geometric or not; write it as a pattern. Why do you think we go over 2 decimal places instead of 1? 12.2 - Geometric Series

  32. Example 12 Write 0.23232323…as a fraction in simplest form Step 2: Find the common ratio 12.2 - Geometric Series

  33. Example 12 Write 0.23232323… as a fraction in simplest form Step 3: Apply the equation Math, ENT, ENT 12.2 - Geometric Series

  34. Example 13 Write 0.123123123… as a fraction in simplest form 12.2 - Geometric Series

  35. Your Turn Write 0.054054…as a fraction in simplest form 12.2 - Geometric Series

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