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Math 152, Benjamin
Aurispa
1. Find the limits of the following sequences if they exist. Does the sequence converge or diverge? Is it
increasing/decreasing? Is it bounded?
(a) an =
n2 + 4
2n2 − 4
(b) an =
(−1)n (n2 + 4)
2n2 − 4
(c) an =
n
n!
(d) an = (−1)n e−n
(e) an = sin(n)
(f) an =
cos(n)
n
(g) a1 = 3, an+1 = 5 −
4
(You are told the sequence is bounded and increasing.)
an
1
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Math 152, Benjamin
Aurispa
2. Do the following series converge, diverge, or do we not know (yet)? If it converges, can we find the
sum of the series?
(a)
∞
X
n
n=1
(b)
∞
X
n+9
n=1
(c)
ln n
3n + 1
∞ X
e−(n+2) − e−n
n=1
2
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Math 152, Benjamin
Aurispa
(d)
∞
X
3n+1
n=1
(e)
∞
X
3n+1
n=0
(f)
(−5)n
(−5)n
∞
X
n=1
n2
3
+n
3
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Math 152, Benjamin
Aurispa
(g)
(h)
∞
X
22n
(−3)n
n=1
∞
X
n
n=1
(i)
∞
X
n=1
3n
√
1
n−5
3. Suppose for a series
∞
X
an it is known that sn = ln(6n + 5) − ln(2n + 1).
n=1
(a) What is the sum of the first 3 terms?
(b) What is the third term of the series?
(c) What is the sum of the series or does it diverge?
(d) What is lim an ?
n→∞
4