Infinite Geometric Series

For r >1, the expressions go to infinity, so there is no limit. For r <-1, the expressions alternate between big positive and big negative numbers, so there is no limit. For r =-1, the expressions alternate between -1 and 1, so there is no limit.

What is an infinite series? An infinite series is a series of numbers that never ends being summed. Example: 1 + 2 + 3 + 4 + 5 + …. Strangely, sometimes infinite series have a finite sum (stops at a number). Other times infinite series sum to an infinitely large number (no sum).

Infinite series can either… Converge – have a finite sum Diverge – keep growing to infinity (no sum)

Infinite GEOMETRIC series… Have a common ratio between terms. Many infinite series are not geometric. We are just going to work with geometric ones.

Example: Does this series have a sum? IMPORTANT! First, we have to see if there even is a sum. We do this by finding r. If | r | < 1, If -1 < r < 1 ) there is a finite sum we CAN find. If | r | ≥ 1, the series sums to infinity (no sum). Let’s find r….

We find r by dividing the second term by the first. In calculator: (1 ÷ 4) ÷ (1 ÷ 2) enter. Absolute value smaller than 1? Has a sum! Now to find the sum…

The sum of an infinite series… Variables: S = sum r = common ratio between terms a1 = first term of series

What did we get as a sum? _____ We found the sum of the infinite series Does this converge or diverge?

You try: 1 – 2 + 4 – 8 + ….. Find the sum (if it exists) of: Remember, fist find r…

We can express infinite geometric sums with Sigma Notation.

Example 2A: Find the Sums of Infinite Geometric Series Find the sum of the infinite geometric series, if it exists. 1 – 0.2 + 0.04 – 0.008 + ... r = –0.2 Converges: |r| < 1. Sum formula

Example 2B: Find the Sums of Infinite Geometric Series Find the sum of the infinite geometric series, if it exists. Evaluate. Converges: |r| < 1.

Check It Out! Example 2a Find the sum of the infinite geometric series, if it exists. r = –0.2 Converges: |r| < 1. Sum formula 125 6 =

Check It Out! Example 2b Find the sum of the infinite geometric series, if it exists. Evaluate. Converges: |r| < 1

You can use infinite series to write a repeating decimal as a fraction.

Example 3: Writing Repeating Decimals as Fractions Write 0.63 as a fraction in simplest form. Step 1 Write the repeating decimal as an infinite geometric series. 0.636363... = 0.63 + 0.0063 + 0.000063 + ... Use the pattern for the series.

Example 3 Continued Step 2 Find the common ratio. |r | < 1; the series converges to a sum.

Check It Out! Example 3 Continued Step 2 Find the common ratio. |r | < 1; the series converges to a sum. Step 3 Find the sum. Apply the sum formula.

You have used series to find the sums of many sets of numbers, such as the first 100 natural numbers. The formulas that you used for such sums can be proved by using a type of mathematical proof called mathematical induction.

Example 4: Proving with Mathematical Induction Use mathematical induction to prove that Step 1 Base case: Show that the statement is true for n = 1. The base case is true. Step 2 Assume that the statement is true for a natural number k. Replace n with k.

Example 4 Continued Add numerators. Simplify. Factor out (k + 1). Write with (k + 1).

Check It Out! Example 4 Use mathematical induction to prove that the sum of the first n odd numbers is 1 + 3 + 5 + … +(2n - 1) = n2. Step 1 Base case: Show that the statement is true for n = 1. (2n – 1) = n2 2(1) – 1 = 12 1 = 12

Check It Out! Example 4 Continued Step 2 Assume that the statement is true for a natural number k. 1 + 3 + … (2k – 1) = k2 Step 3 Prove that it is true for the natural number k + 1. 1 + 3 + … (2k – 1) + [2(k + 1) – 1] = k2 + [2(k + 1) – 1] = k2 + 2k + 1 = (k + 1)2 Therefore, 1 + 3 + 5 + … + (2n – 1) = n2.

Evaluate: