SERIES AND CONVERGENCE Chapter 9.2 SERIES AND CONVERGENCE

After you finish your HOMEWORK you will be able to… Understand the definition of a convergent infinite series Use properties of infinite geometric series Use the nth-Term Test for Divergence of an infinite series

INFINITE SERIES An infinite series (aka series) is the sum of the terms of an infinite sequence. Each of the numbers, , are called terms of the series.

CONVERGENT AND DIVERGENT SERIES For the infinite series , the n-th partial sum is given by . If the sequence of partial sums, , converges to , then the series converges. The limit is called the sum of the series. If diverges, then the series diverges. Series may also start with n = 0.

THE BATHTUB ANALOGY

DIVERGE VERSUS CONVERGE Consider the series What happens if you continue adding 1 cup of water? Consider the series How is this situation different? Will the tub fill?

TELESCOPING SERIES What do you notice about the following series? What is the nth partial sum?

CONVERGENCE OF A TELESCOPING SERIES A telescoping series will converge if and only if approaches a finite number as n approaches infinity. If it does converge, its sum is

GEOMETRIC SERIES The following series is a geometric series with ratio r.

THEOREM 9.6 CONVERGENCE OF A GEOMETRIC SERIES A geometric series with ratio r diverges if . If then the series converges to

THEOREM 9.7 PROPERTIES OF INFINITE SERIES If is a real number, then the following series converge to the indicated sums.

THEOREM 9.8 LIMIT OF nth TERM OF A CONVERGENT SERIES If converges, then Why?

The nth-Term Test THEOREM 9.9 If , the infinite series diverges.