Limits

An infinite geometric series has infinitely many terms An infinite geometric series has infinitely many terms. Consider the two infinite geometric series below.

Notice that the series Sn has a common ratio of and the partial sums get closer and closer to 1 as n increases. When |r|< 1 and the partial sum approaches a fixed number, the series is said to converge. The number that the partial sums approach, as n increases, is called a limit.

For the series Rn, the opposite applies For the series Rn, the opposite applies. Its common ratio is 2, and its partial sums increase toward infinity. When |r| ≥ 1 and the partial sum does not approach a fixed number, the series is said to diverge.

Example 1: Finding Convergent or Divergent Series Determine whether each geometric series converges or diverges. A. 10 + 1 + 0.1 + 0.01 + ... B. 4 + 12 + 36 + 108 + ... The series converges and has a sum. The series diverges and does not have a sum.

If an infinite series converges, we can find the sum If an infinite series converges, we can find the sum. Consider the series from the previous page. Use the formula for the partial sum of a geometric series with and

Find the sum of the infinite geometric series, if it exists. 1 – 0.2 + 0.04 – 0.008 + ...

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.

Another application of infinite sequences… 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.

 Example 3 Continued Step 3 Find the sum. Apply the sum formula. Check Use a calculator to divide the fraction

There are times when it is helpful to examine the limits of sequences as well…we want to see what is going on out to infinity and beyond! The idea of limits can apply to patterns other than geometric patterns.

We know some limits already that can help us on our exploration… Lim n ∞ 1/n = ?

When we have equations that involve more than just that limit identity, we need some properties to help us actually find the limit. We will take complicated items and break it into manageable parts using the following:

Lesson Overview 12-3A

Two ways to attack one of these problems… 1. Simplify into parts…

If simplifying is not an option, then divide by the highest power of n to form an equivalent form.

 What about this one…

Lesson Overview 12-3B

5-Minute Check Lesson 12-4A

5-Minute Check Lesson 12-4B