12. Section 10.2 Summing an Infinite Series Continued

Section 10.2 Summing an Infinite Series Continued EQ – What other types of series (besides geometric) are known to converge or diverge?

Yesterday Yesterday we saw that geometric series converge and we can find a definite sum. Today we will look at other types of series.

Telescoping series Another kind of series that we can sum: telescoping series A series is said to telescope if all the terms in the partial sums cancel except for the first and the last.

Finding sums of telescoping series Can use partial fractions to rewrite

Now you try one... A) 1 B) 3/4 C) 1/2 D) 1/4 E) 1/8

The convergence question One property that convergent series must have is that their terms must get smaller and smaller in order for the limit of the partial sums to exist.

Fundamental necessary condition for convergence: Careful!! This is only a test you can use to prove that a series does NOT converge, it does NOT prove that it DOES converge nth term test for divergence: If the limit as n goes to infinity for the nth term is not 0, the series DIVERGES!

Does the series diverge? So, the series diverges by the Test for Divergence. TEST FOR DIVERGENCE

Another Example Does converge? Can’t tell

Harmonic Series Just because the nth term goes to zero doesn't mean that the series converges. An important example is the harmonic series We can show that the harmonic series diverges by the following argument using the partial sums: For the harmonic series,

Harmonic (cont.) and so on -- every time we double the number of terms, we add at least one more half. This indicates that

Whenever we have convergent series, we can add them, subtract them, or multiply them by a constant, and the resultant series are also convergent Properties of Series

Evaluate. Example

Series known to converge or diverge 1. A geometric series with | r | <1 converges 3. A repeating decimal converges 4. A telescoping series converges 2. A geometric series with | r | >1 diverges 5. A harmonic series diverges 6. If series diverges, (can’t tell about convergence)

Assignment Pg. 571: # 9-17 odd, 40, 41