5 Steps to a 5 AP Calculus BC 2019

348 STEP 4. Review the Knowledge You Need to Score High

Example 1

Determine whether the series

∑∞

n= 1

1

5 n

converges or diverges. If it converges, find its sum.

Step 1: Find the first few partial sums.

s 1 =

1

5

= 0 .2,s 2 =

1

5

+

1

25

=

6

25

= 0. 24 s 3 =

1

5

+

1

25

+

1

125

=

31

125

= 0. 248

s 4 =

1

5

+

1

25

+

1

125

+

1

625

=

156

625

= 0. 2496

Step 2: The sequence of partial sums{ 0 .2, 0.24, 0.248, 0.2496,...}converges to 0.25,

so the series converges, and its sum

∑∞

n= 1

1

5 n

= 0 .25.

Example 2

Find the sum of the series

∑∞

n= 1

(5an− 3 bn), given that

∑∞

n= 1

an=4 and

∑∞

n= 1

bn=8.

Step 1:

∑∞

n= 1

(5an− 3 bn)=

∑∞

n= 1

5 an−

∑∞

n= 1

3 bn= 5

∑∞

n= 1

an− 3

∑∞

n= 1

bn

Step 2: 5

∑∞

n= 1

an− 3

∑∞

n= 2

bn=5(4)−3(8)= 20 − 24 =− 4

14.2 Types of Series

Main Concepts:p-Series, Harmonic Series, Geometric Series, Decimal Expansion

p-Series

Thep-series is a series of the form 1+

1

2 p

+

1

3 p

+

1

4 p

+···+

1

np

+···=

∑∞

n= 1

1

np

. The

p-series converges whenp>1, and diverges when 0<p≤1.

Harmonic Series

The harmonic series 1+

1

2

+

1

3

+

1

4

+···+

1

n

+···=

∑∞

n= 1

1

n

is ap-series withp=1. The

harmonic series diverges.

Geometric Series

A geometric series is a series of the form

∑∞

n= 1

arn−^1 wherea=/0. A geometric series converges

when|r|<1. The sum of the firstnterms of a geometric series issn=

a(1−rn)

1 −r

. The sum

of the series

∑∞

n= 1

arn−^1 =nlim→∞sn=nlim→∞

a(1−rn)

1 −r

=

a

1 −r

.