The sum of an Infinite Series 10.2B The sum of an Infinite Series

Then the sum of the series is: A finite series is given by the terms of a finite sequence, added together. For example, we could take the finite sequence (3,5,7,…, 21) 𝑎 𝑛 =2𝑛+1 , 1≤𝑛 ≤10 Then the corresponding example of a finite series would be given by all of these terms added together, 3+5+7+…+21 Then the sum of the series is: =120

An infinite series is the sum of the terms of an infinite sequence. 𝑎 𝑛 = 1 2 𝑛 , 𝑛 ≥1 ( 1 2 , 1 4 , 1 8 , …) and then the series obtained from this sequence would be 1 2 + 1 4 + 1 8 +… 𝑛=1 ∞ 1 2 𝑛 use sigma notation to express this series

A finite series is given by all the terms of a finite sequence, added together. An infinite series is given by all the terms of an infinite sequence, added together.

The sum of an infinite series 𝑛=1 ∞ 1 2 𝑛 The sum of an infinite series Let’s use Let us try adding up the first few terms and see what happens. These sums of the first terms of the series are called partial sums If we add up the first two terms we get 1 2 + 1 4 = 3 4 The sum of the first three terms is 1 2 + 1 4 + 1 8 = 7 8 The n-th term of this sequence is the n-th partial sum 1 2 + 1 4 + 1 8 + 1 16 = 15 16 The sum of the first four terms is

These partial sums appear to get closer to 1 These partial sums appear to get closer to 1. An estimate of the finite sum is 1. In general, we say that an infinite series has a sum if the partial sums form a sequence that has a real limit. We can see that the partial sums here form a sequence that has limit 1. So it would make sense to say that this series has sum 1

Predict whether each infinite geometric series has a finite sum Predict whether each infinite geometric series has a finite sum. Estimate each finite sum. 𝑠 1= 0.5 𝑠 2 = 𝑠 1 +1 𝑠 3 = 𝑠 2 +2 𝑠 4 = 𝑠 3 +4 7.5 1.5 3.5 As the number of terms increases, the partial sums increase, so the series does not have a finite sum.

What is r ? The common ratio is not between -1 and 1, so the series diverges.