AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606

INFINITE SERIES

 If an is an infinite sequence, the aaaaann 1234... a ... is an infinite series. (For n1 some series is convenient to begin the index at n=0)

To find the sum of an infinite series, consider the following sequence of partial sums.

Sa11

Saa212

Saaa3123 ...

Saaann123... a If this sequence of partial sums converges, the series is said to converge and has the sum indicated in the following definition.

AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606

Sample problem #1: CONVERGENT AND DIVERGENT SERIES

Determine if the given series is convergent or divergent.

 3 a)  3 n2 52n 

AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606

n  4 b)  n1 3 AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606

 11 c)  n1 nn1

The series in (c) is a telescoping series of the form

bb12   bb 23  bb 34  bb 45 ...

Note that all b(s), from b2 to bn , are canceled, therefore in a telescoping series

Sbbnn 11 

It follows that a telescoping series will

converge if and only if bn approaches a finite number as n . Moreover, if the series converges, its sum is

Sb 11 lim bn n

AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606

Sample problem #2: WRITING SERIES IN TELESCOPING FORM

 3 Determine the sum of the series  2 n1 12n  2

AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606

GEOMETRIC SERIES  A geometric series has the form arnn a ara  r 2 ... ar .. . a  0 (ratio: r) n0

Sample problem #3: CONVERGENT AND DIVERGENT GEOMETRIC SERIES

Determine, if possible, the sum of the given series.

 5  5 a) b)  n  n n2 8 n0 3

AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606 c) A GEOMETRIC SERIES FOR A REPEATING DECIMAL

Use a geometric series to write 0.05 as the ratio of two integers.

AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606

AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606

Nth –TERM TEST FOR DIVERGENCE

The following theorem states that if a series converges, the limit of its nth term must be 0.

The contrapositive of the above theorem provides a useful test for divergence. This nth-Term for Divergence states that is the limit of the nth term of a series does not converge to 0, the series must diverge.

AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606

Sample problem #3: USING THE nth-TERM TEST FOR DIVERGENCE

Determine, if possible, if the series diverges based on the nth-Term for Divergence.

   n n 2!n 1 a) 5 b)  c)  n0 n0 13! n n0 5