Name:
Math 1220
Exam 4 Review
1. For each of the following sequences:
• Determine if the sequence converges
• If it converges, find the limit of the sequence as n → ∞
(a) an =
n3 + ln(n)
(n + 1)3
(b) an = e−n sin(n)
(c) 1,
1
1
1
,
,
,...
1
2
1 − 2 1 − 3 1 − 34
2. Determine if the following series converge. If the series converges, find its sum.
(a)
∞
X
2n
3n+1
(b)
∞
X
n=1
n=1
1
n(n + 1)
3. Determine if the following series converge or diverge. Be sure to justify your conclusion and specify
what tests you used.
(a)
∞
X
n2 + 5
n=1
(b)
∞
X
n2 − 5
ne−n
2
(c)
n=1
(d)
n=1
∞
X
(ln(n))2
(e)
n
n=1
√
∞
X
n+2 n
n=1
∞
X
n2
(f)
n3 − 6
n!
∞
X
1 + cos2 (n)
n=1
n2
4. Determine if the following series converge conditionally, converge absolutely, or diverge. Be sure to
justify your conclusion and specify what tests you used.
(a)
∞
X
n=1
(b)
∞
X
n=1
∞
X
1
(c)
(−1)n √
2
n +1
n=1
1
(−1)
n ln n
n
(−1)n
n3
n!
(d)
∞
X
sin(πn)
n=1
n2
5. Determine the error in using the partial sum S4 to approximate the sum of the following series:
∞
X
(−1)n
n=1
1
n2 + 1
How many terms would you need to sum in order for the error to be less than 0.1?
6. For what values of x does the following series converge?
∞ n
X
x
n=0
2n