1
10.2: Series
Definitions:
Infinite Series:
N th Partial Sum:
Convergent Series:
Divergent Series:
Special (Summable) Kinds of Series:
Geometric Series:
Find the values of r for which
∞
X
arn−1 is convergent and find the sum.
n=1
1
Telescoping Series:
Find the sum of
∞ X
1
n=1
n
−
1
n+2
Properties of Convergent Series:
If
∞
X
an and
n=0
i)
∞
X
bn are convergent, then:
n=0
ii)
iii)
Tests for Convergence of Series: (continued through 10.4)
I. The Test for Divergence (or Divergence Test):
∞
X
n+3
2n + 1
n=0
2
.
On Beyond Average:
A window consists of two parallel panes of tinted glass with a small air space between the panes.
Whenever light hits either pane of glass on either side, 1/3 of the light passes through the pane and
2/3 is reflected back. If a light shines on one side of the window, what fraction of the light passes
through the window?
The N th partial sum of a series is given by sN =
the series is divergent.
N +1
. Find the sum of the series or explain why
2N + 3
Prove the converse of the Test for Divergence is false by showing that
1
→ 0).
n
3
∞
X
1
diverges (even though
n
n=1