Math 132 - Calculus II
Loyola Marymount University
Spring 2017
§9.1
§9.4
• Notation
• Divergence Test (nth Term Test)
⋄ Indexing
⋄ lim ak
⋄ Sequences vs Series
⋄ Conditions for Conv, Div, or Inconclusive
k→∞
• Sequences
⋄ Recurrence Relation / Implicit Formula
⋄ Explicit Formula
• Series
⋄ Partial Sums
⋄ Infinite Series
a
⋄ Conditions for Application
⋄ Conditions for Conv, Div, or Inconclusive
• p-Series
⋄
• Limits
⋄ lim an vs. lim Sn
n→∞
• Integral Test
ˆ ∞
f (x) dx
⋄
n→∞
§9.2
• Limit of Sequences
⋄ Properties of Limits and Techniques
• Monotonic Sequences
⋄ Nondecreasing
⋄ Nonincreasing
• Bounded Monotonic Sequences
• Geometric Sequences
⋄ Conditions for Conv. / Div.
• Squeeze Theorem for Sequences
§9.3
• Infinite Series
• Geometric Series
1 − rn
⋄ Sn = a ·
1−r
a
⋄ S∞ =
, |r| < 1
1−r
• Telescoping Series
X 1
kp
⋄ Conditions for Application
⋄ Conditions for Conv, Div, or Inconclusive
• Remainder
⋄ Rn = |S − Sn |
• Remainder for General Series with Pos. Terms
ˆ ∞
ˆ ∞
f (x) dx
f (x) dx ≤ Rn ≤
⋄
n
n+1
• Bounds for General Series with Pos. Terms
ˆ∞
ˆ∞
⋄ Sn +
f (x) dx ≤ S ≤ Sn + f (x) dx
n+1
n
• Properties of Convergent Series
§9.5
• Ratio Test
ak+1
⋄ lim
k→∞ ak
⋄ Conditions for Application
⋄ Conditions for Conv, Div, or Inconclusive
• Root Test
√
⋄ lim k ak
k→∞
⋄ Partial Fraction Decomposition
⋄ Conditions for Application
⋄ Solve for Sn and S∞
⋄ Conditions for Conv, Div, or Inconclusive
Math 132 - Calculus II
Loyola Marymount University
• Comparison Test
Example 5:
(
⋄ Conditions for Conv, Div, or Inconclusive
• Limit Comparison Test
n
Example 6:
ak
k→∞ bk
⋄ lim
Find the limit of the sequence
⋄ Conditions for Application
(−1)n n
.
n + 81
Example 7:
⋄ Conditions for Conv, Div, or Inconclusive
The continued fraction shown below can be expressed
by the implicit formula, a0 = 1, an+1 = 1 + 1/an .
Find the value of the infinite fraction.
§9.6
• Alternating Series
X
)
7n+1
Find the limit of the sequence cos (0.999 ) + n .
5
⋄ Conditions for Application
⋄
Spring 2017
1
1+
(−1)k+1 ak
1
1+
1
1+
• Alternating Series Test
1+
⋄ lim ak
k→∞
1
..
.
⋄ Conditions for Application
Example 8:
⋄ Conditions for Conv, Div, or Inconclusive
Determine if the series converges
• Remainder in Alternating Series
⋄ Rn ≤ an+1
∞
X
1
.
k ln k ln (ln k)
k=1
Example 9:
• Absolute Conv. vs Conditional Conv.
Determine if the series converges
∞
X
n sin3 (6n)
n+1
n=1
Example 1:
.
Example 10:
Find a recurrence relation andnan √
explicit formula
for
o
th
the n term of the sequence. 1, 2 , 2, . . .
Example 2:
Determine if the series converges
∞
X
n=1
√
k3
.
9 + 4k5
Example 11:
For the following infinite series write an explicit
formula for the nth partial sum.
Determine if the series converges
∞ k
X
2 + 3k
5k
k=1
4 + 0.7 + 0.007 + 0.0007 + · · ·
.
Example 12:
Example 3:
Find the limit of the sequence
( s
1+
1
2n
n )
Determine if the series converges
∞
X
k=1
.
1
√ √
.
k k+1
Example 13:
Example 4:
Find the limit of the sequence {ln sin (1/n) + ln n}.
Determine if the series converges
∞
X
k=1
√
k
k
k3
.
Math 132 - Calculus II
Loyola Marymount University
Example 14:
Determine if the series converges
∞
X
k=1
√
k
k
.
k3
Example 15:
Determine if the series converges
∞
X
k sin k.
k=1
Example 16:
Determine if the series converges
∞ k
X
2 k!
k=1
kk
.
Example 17:
∞
X
k+2
.
ln
Determine if the series converges
k+1
k=1
Example 18:
Determine if the series converges
∞
X
sin (1/k)
k=1
k2
.
Example 19:
Determine if the series converges
∞
X
1
√ .
2k
−
k
k=1
Example 20:
Sierpinski’s triangle exercise Review section #67
Spring 2017