Review Write an explicit formula for the following sequences. 5, 7, 9, 11, 13, 15. . . 1, 3, 9, 27, 81, 243. . .

Answers Write an explicit formula for the following sequences. 5, 7, 9, 11, 13, 15. . . an = 2n + 3 2. 1, 3, 9, 27, 81, 243. . . an = 1 • 3 (n-1)

Sum of numbers 1-100?

Series and Summation Notation

Series A series is the sum of the terms in a sequence. 3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 = 135 3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 = 135

a1•(1 – rn) n ________ Sn = – (a1 + an) 2 Sn = (1 – r) Series Formulas Arithmetic Series Geometric Series n 2 a1•(1 – rn) (1 – r) ________ Sn = – (a1 + an) Sn =

Steps for Evaluating a Series Determine if the terms in the series are arithmetic or geometric. Common difference or common ratio? Substitute the information you are given into the appropriate formula.

Example 1 Evaluate a series with the terms 1, 7, 13, 19, 25 for the first 13 terms.

Example 2 Find the sum of the first 10 terms of the geometric series with a1 = 6 and r = 2.

Example 3 A philanthropist donates $50 to the SPCA. Each year, he pledges to donate 12 dollars more than the previous year. In 8 years, what is the total amount he will have donated?

I’m just a fancy way of saying, “Add everything up!” Summation Notation Instead of saying: “Find the sum of the series denoted by an = 3n + 2 from the 3rd term to 7th term,” mathematicians made up a symbol to deal with it. Sigma! ∑ I’m just a fancy way of saying, “Add everything up!”

Summation Notation last term sequence formula first term

Example 4: Evaluating Using Summation Notation

Writing Series in Summation Notation Step 1) Determine the explicit formula. Step 2) Identify the lower and upper limits. Step 3) Write the series in summation notation.

Example 5 Use summation notation to write the series for the specified number of terms. 1 + 2 + 3 + …; n = 6

Example 6 Use summation notation to write the series for the specified number of terms. 3 + 8 + 13 + 18 + …; n = 9

Example 7 Use summation notation to write the series for the specified number of terms. 3 + 6 + 9 + …; n = 33

Example 8 Use summation notation to write the series for the specified number of terms. 8 + 4 + 2 + …; n = 12

Infinite vs. Finite The difference between a finite and infinite series is whether or not there is a “…” at the end. Example, 3, 5, 7, 9…. 6, 3, 1.5, .75

Convergent and Divergent Series

What will happen to the terms of this sequences as it continues forever? It approaches 0!

Convergent Sequence A sequence is converging if its terms approach 0. *Only applies to geometric sequences.

Determine if the following sequences are convergent or divergent:

Sum of an Infinite Geometric Series We can find the sum of this series, even though it goes on forever, because it is convergent.

Sum of an Infinite Geometric Series The following conditions must be true to use this formula: The series must be geometric The series must be convergent The series must be infinite

Example 1 Determine whether each infinite geometric series diverges or converges. If it converges, find the sum. a) 48 + 12 + 3 + … b) 4 + 8 + 16 + …

Example 2 Evaluate each infinite geometric series a) b)

Formula Recap Sequence Series Arithmetic Geometric Finite Infinite

Given 4, 20, 100, 500, … (a) find the explicit formula (b) find the 15th term (c) find the sum of the first 15 numbers