13.2 Series

• Sequence2, 4, 6, …, 2n, …

• A sequence is a list.

• Related finite series2 + 4 + 6 + 8

• Related infinite series2 + 4 + 6 + … + 2n + …

• Series is the list as a sum.

• Every finite series represents a partial sum of a sequence . • Partial sum is the total of the terms of a sequence up to a given

term.

nS na

• ExampleIf , what is the sum of the finite series formed by adding the first five terms of the sequence ?

2n

na

5S

na

• Find for each of the following infinite series. If the sequence were to continue indefinitely, does it seem to have a limit or not? If yes, predict the limit.

#1 1+2+4+8+16+…#2 0.3+0.03+0.003+0.0003+0.00003+…#3 1+0.5+(0.5)^2+(0.5)^3+(0.5)^4+…

1 2 3 4 5, , , ,S S S S S

• Sigma notation is useful for describing series.

• If the graph of a sequence of partial sums does not approach a horizontal asymptote, then we say that the infinite series does not have a sum.

Summary• An infinite sequence can have a limit. This is the value that

the sequence’s terms approach as the index goes to infinity.

• An infinite series can converge or diverge. If the series converges, it has a specific sum; otherwise it does not.

• These two concepts are related. Associated with any infinite series is an infinite sequence of partial sums.

• If the infinite sequence of partial sums has a limit, the series is said to converge, with a sum equal to the limit.

• If the infinite sequence of partial sums does not have a limit, the series to said to diverge and has no sum.

• ExampleEvaluate

• ExampleEvaluate

6

1

(2 1)k

k

1

1

13 ( )

2n

n

• ExampleEvaluate

• ExampleEvaluate

1

118( )10

n

n

1n

n