Infinite Series - Section 10.2 Can you add up an infinite number of values and get a finite sum? Yes! Here is a familiar example: 1 3 0. 3 0. 3 0. 03 0. 003 0. 0003 Ifa n is an infinite sequence, then a n a 1 a 2 a 3 a n n1 is called an infinite series (or just a series). The a 1, a 2, a 3, are called the terms of the series. For some series it is convenient to begin the index at n 0 (or some other integer). It is common to represent a series simply as a n. The partial sums of a series are s 1 a 1 s 2 a 1 a 2 s 3 a 1 a 2 a 3 n s n a 1 a 2 a 3 a n a i i1 Note that these partial sums form an infinite sequence s n that we call the sequence of partial sums. Definition for convergent and divergent series: If the sequence of partial sumss n of a series converges to S (i.e. lim n s n S), then the series a n converges. The limit S is called the sum of the series and we write a n S nk If lim n s n does not exist, then a n diverges. If lim n s n is infinite ( or ), then we say a n diverges to infinity. Find the sum of the following infinite series and explain your reasoning: 1. 0. 1 0. 01 0. 001 0. 0001
The Geometric Series: The geometric series with ratio r is a arar 2 ar 3 ar n1 ar n ar n n0 This series diverges if r 1. If r 1, then the series converges to the sum n0 ar n ar n1 a 1 r n1 r 1 Proof:
Geometric Series examples: 1. Is the series convergent or divergent? If it is convergent, what is its sum? a. n0 3 2 n b. n2 3 2 n c. n0 3 2 n d. n1 2 2n 3 n1
2. Use the geometric series to express 0. 080808 as the ratio of two integers. Telescoping Series: A telescoping series is of the form b 1 b 2 b 2 b 3 b 3 b 4 b 4 b 5 Note that the nth partial sum is s n b 1 b n1. Thus a telescoping series will only converge ifb n approaches a finite limit. In this case, the series is convergent and the sum is S n lim s n b 1 n lim b n1. Find the sum of the following series: n1 1 n 1 n 1
An Important Convergence Theorem: If Proof: n1 a n converges, then lim n a n 0. Note that the converse of this theorem (if lim n a n 0, then a n converges) is NOT true. But, the contrapositive of this theorem is true and yields the following useful theorem: The nth Term Test for Divergence: If n lim a n 0, then a n diverges. n1 n1 Apply the nth Term Test for Divergence to the following series: 1. cos n1 1 n 2. n1 n 2 5 3n 2 4n 2 3. 1n n1
The Harmonic Series: The series 1n is called the Harmonic Sequence. It is divergent. We will prove this n1 later. The following properties of series are a direct consequence of the corresponding properties of sequences. If a n A, b n B, and c is a real number then the following series converge to the indicated sums n1 ca n c a n ca n1 a n b n A B n1 a n b n A B n1 Use these properties to find the sums of the infinite series n0 6 4 n 5 n
An application: A ball with perfectly elastic bounces A ball is dropped from a height of 6 feet and begins to bounce. The height of each bounce is three-fourths of the height of the previous bounce. What is the total vertical distance traveled by the ball.
Consider the series 2 n n! n0 List the first four terms that are added up in this series. Find the first 4 terms of the sequence of partial sums for this series.
Suppose that S a n is a convergent series with partial sum s n 5 2 n 2 n1 5 What is the value of a n? n1 What is the value of a 3? Find a general formula for a n Find the sum a n n1 What is lim n a n?