Finding the sum of a finite Geometric Series. The sum of the first 5 powers of 2 The sum of the first 5 powers of 3

Transcription

1 Section 1 3B: Series A series is the sum of a given number of terms in a sequence. For every sequence a 1, a, a 3, a 4, a 5, a 6, a 7,..., a n of real numbers there is a series that is defined as the sum of the terms of the sequence a 1 + a + a 3 + a 4 + a 5 + a 6 + a a n If the sequence ends after a given number of terms a 1, a, a 3, a 4, a 5, a 6 then the series it is based on will also end after that number of terms a 1 + a + a 3 + a 4 + a 5 + a 6. A series that has a fixed or finite number of terms is called a finite series. If the sequence has an infinite number of terms a 1, a, a 3, a 4, a 5, a 6,... then the series it is based on will also have an infinite number of terms a 1 + a + a 3 + a 4 + a 5 + a A series that has an infinite number of terms is called a infinite series. If the sequence is an geometric sequence then the series the sequence is based on will be a Series. Some of the information from section 1 about series will apply in this section but additional formulas can be used if the series is geometric. Finding the sum of a finite Series. The sum of the first 5 powers of The sum of the first 5 powers of = = 363 If there are a small number of terms the sum is easy to find. As the number of terms increase we need start to look for a formula that will help simplify the work. The Sum of the First n terms of an Sequence For a Sequence whose first term is a 1 and whose common ratio is r where r 0,1, 1 the sum S n of the first n terms of is n S n = a 1 r k 1 = a 1 1 ( r)n k =1 1 ( r) Section 1 3B Page Eitel

2 Example 1 Find the sum of of the first 1 terms of r = 3 16 = r = 16 8 = r = 8 4 = a 1 = 4 r = n = 1 S 1 = a 1 1 ( r)n 1 r ( ) S 1 = 4 1 ( )1 1 ( ) 1 1 = 4 1 S 1 = 4 ( 1 1 ) If a calculator is not allowed then leave the answer with the exponent Using a calculator S 1 = 16,380 Note: If the use of a calculator is not allowed then you should leave the answer in exponential form Section 1 3B Page 015 Eitel

3 Example Find the sum of of the first 7 terms of r = 81 7 = 3 r = 7 9 = 3 r = 9 3 = 3 r = 3 1 = 3 a 1 =1 r = 3 n = 7 S 7 = a 1 1 ( r)n 1 r 1 ( 3)7 S 7 = 1 1 ( 3) 1 ( 3)7 =1 4 1 ( 3)7 S 7 = 1 4 = Note: The calculator rounds off values involving large exponents. In many cases large exponents will result in the calculator rounding the answer and using scientific notation. The resulting answer is not exact. Leaving the answer in exponential form gives the exact answer. Section 1 3B Page Eitel

4 Example 3 Find the sum of of the first 10 terms of r = 1 = 1 r = 4 = 1 r = 4 8 = 1 r = 8 16 = 1 a 1 =16 r = 1 3 n = 10 S 10 = a 1 1 ( r)n 1 r S 10 = =16 1 S 10 = k=1 4 ( 3 k 1 ) Example 4 Find the sum of ( 3 11 ) r = = 3 r = 36 1 = 3 r = 1 4 = 3 Since the sequence starts with k = 1 the upper index states the last term is the 1th term a 1 = 4 r = 3 n = 1 S 1 = a 1 1 ( r)n 1 r S 1 = 4 1 ( 3)1 1 3 ( ) S 1 = [ 1 ( 3) 1 ] = 4 1 ( 3)1 Section 1 3B Page Eitel

5 Infinite Series An infinite Series can be written as a 1 + a 1 r + a 1 r a 1 r n this series comtinues without end for values of n =1,,3,4..., and is denoted by an upper index of a 1 r n 1 k=1 Can you find the sum of an Series if there are an infinite number of terms? That depends on the value of the common ratio r. If r is between 1 and 1 we can write this as r <1. If r <1 then the infinite number of terms will have an exact sum. We say the series converges to that sum. If r is not between 1 and 1 then the sum of the infinite terms increase to + or decreases to without bound. We say the series diverges. There is no sum for the infinite number of terms. Determine whether a Series Converges or Diverges If r < 1 then the infinite Series converges. The sum of the infinite number of terms is given by a 1 r k 1 = a 1 k=1 1 r If the Series does NOT converge then it diverges Section 1 3B Page Eitel

6 Example 1 k 1 1 k=1 = = a 1 =1 r = 1 r <1 so the series converges to a 1 1 (r) = = 1 1 = 1 k 1 1 = k=1 Note: This results says that the total of the infinite number of fractions is EXACTLY. The partial sums get closer and closer to 6 but the sum of the infinite number of term is exactly 6. This concept will be explained further in your calculus course. One of the major topics in that course is the limits of infinite sums. S 1 =1 S = = 3 =11 S 3 = = S 4 = = 15 8 = 17 8 S 5 = = = S 6 = = 63 3 = S 7 = = = S = Section 1 3B Page Eitel

7 Example k 1 k=1 3 = = a 1 = r = 3 r <1 so the series converges to a 1 1 (r) = 1 3 = 1 3 = 3 k 1 k=1 3 = 6 Example k 1 k=1 3 = = a 1 = 9 r = 1 3 r <1 so the series converges to a 1 1 (r) = = 9 3 = n 1 k=1 3 = 7 = 14.5 Section 1 3B Page Eitel

8 Example 4 k 1 1 = k= = a 1 =1 r = 1 r <1 so the series converges to a 1 1 (r) = = 1 = k 1 1 = k=1 3 Section 1 3B Page Eitel

9 Possible sequences if the first term is a 1 positive First Term R Type of Sequence Example a 1 is positive R > 1 Increasing Divergent a 1 =1 and r = a n =1,, 4, 8, 16, 3,... a 1 is positive 0 < R < 1 Decreasing Convergent a 1 = 4 and r = 1/ a n = 4,, 1, 1, 1 4,... a 1 is positive 1 < R < 0 Convergent a 1 = 3 and r = 1/ 3 a n = 3, 1, 1 3, 1 9, 1 7, 1 81,... a 1 is positive 1 < R Divergent a 1 = 3 and r = a n = 3, 6, 1, 4, 48, 96,... Possible sequences if the first term is a 1 negative First Term R Type of Sequence Example a 1 is negative R > 1 Divergent a 1 = and r = 3 a n =, 6, 18, 54, 16,... a 1 is negative 0 < R < 1 Decreasing Convergent a 1 = 4 and r = 1/ a n = 4,, 1, 1, 1 4,... a 1 is negative 1 < R < 0 Convergent a 1 = 4 and r = 1 / a n = 4,, 1, 1, 1 4,... a 1 is negative 1 < R Divergent a 1 = 3 and r = a n = 3, 6, 1, 4, 48, 96,... Section 1 3B Page Eitel