Chapter 13: Sequences & Series

1. Chapter 13: Sequences & Series L13.5 Sums of Infinite Series

2. DO NOW Use your calculator to find S1, S2, S3, S4 and S5 for each of the following infinite series. If this sequence were to continue indefinitely, does it seem to have a limit or not? If yes, predict the limit. • 1. 1 + 2 + 4 + 8 + 16 + … • 0.3 + 0.03 + 0.003 + 0.0003 + 0.00003 + …. • 1 + 0.5 + (0.5)2 + (0.5)3 + (0.5)4 + … 1, 3, 7, 15, 31; No 0.3, 0.33, 0.333, 0.3333, 0.33333; Yes, ⅓ 1, 1.5, 1.75, 1.875, 1.9375; Yes, 2 The values S1, S2, S3, S4 and S5 are called partial sums.

3. Sum of an Infinite Series The sum of an infinite series is very closely connected to the limit of an infinite sequence. Consider the geometric series: Associated with this series are the following partial sums: … The infinite sequence of partial sums of the series is: Which has limit 1. Since the limit of the partial sums is 1, we say the infinite series has limit 1 or has the sum 1.

4. Sum of an Infinite Series If an infinite series has a sequence of partial sums with a finite limit, S, then the series is said to converge to the sum, S. If an infinite series’ sequence of partial sums approaches infinity or has no finite limit, then the infinite series is said to diverge. Consider the infinite geometric series: t1 + t1r + t1r2 + t1r3 + … + t1rn + … Its nth partial sum is If |r| < 1, the term rn approaches 0 as n→∞, and the sum of the infinite geometric series converges to the sum If |r| ≥1 and t1 ≠ 0, then the series diverges and has no sum. This is a special case!! If the series is not geometric, you must take the limit of the sequence of partial sums.

5. Examples • Find the sum of 24 – 12 + 6 – 3 + …. • Find the sum of 1 + 3 + 9 + 27 + 81 + … • Find the first 3 terms of the infinite geometric series with sum ⅓ and common ratio −½. This is a geometric series with r = -½. Since |r| < 1, this series converges to the sum This is a geometric series with r = 3. Since |r| ≥ 1, this series diverges and has no sum. Solve for t1: The first 3 terms are then: ½ + -1/4 + 1/8

6. Intervals of Convergence • For what values of x does the following infinite series converge? 1 + x + x2 + x3 + … • Express the sum of this infinite series in terms of x. • For what values of y does the following infinite series converge? 1 + 3y + 9y2 + 27y3 + … Express the sum in terms of y. This is a geometric series with r = x. Since |x| < 1 in order for this series to converge, -1 < x < 1. This is called the interval of convergence ←You do this one

7. Repeating Decimals Show that 0.5 = 5/9. Infinite geometric series can be used to convert repeating decimals to fraction form. Example: 0.45 can be written as 0.45 + 0.0045 + 0.000045 + 0.00000045 + …. What is the sum of this series? t1 = 0.45, r = 0.01 so t1 = 0.5, r = 0.1 so

8. Find the Sum of this Infinite Series Example: This series is not geometric, so you cannot use the formula for convergent geometric series. Instead you must take the limit of the sequence of partial sums. 1. Sequence of partial sums: S1, S2, S3, S4, … is: 2. Find its limit by suggesting a formula for this sequence: 3. And taking limit: = 1 If an infinite series has a sequence of partial sums with a finite limit, S, then the series is said to converge to the sum, S. So the sum of the series above is 1. Note: If the sequence of partial sums does not have a limit, the series is said to diverge and has no sum.

9. SUMMARY An infinite sequence can . This is the value that the sequence’s terms approach as the index goes to infinity. have a limit If the series converges, it has a specific sum; otherwise it does not. An infinite series can . converge or diverge These two concepts are related. Associated with any infinite series is an infinite sequence of partial sums. • If the infinite sequence of partial sums has a limit, the series is said to converge, with a sum equal to the limit. • If the infinite sequence of partial sums does not have a limit, the series is said to diverge and has no sum.