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1. (a) Let (sn)
∞
n=1 and (tn)
∞
n=1 be non-negative bounded sequences. Prove that
lim sup(sntn) ≤ (lim sup sn)(lim sup tn) (Hint: Observe that for any fixed
N ∈ N, for all n > N we have sn ≤ sup{sk|k > N}. Same thing happens
for tn and hence sntn ≤ sup{sk|k > N} sup{tk|k > N}. Use this to prove
the result)
(b) Give an example of non-negative bounded sequences with lim sup(sntn) <
(lim sup sn)(lim sup tn), i.e. the equality does not hold.
2. Determine whether the following series converges or diverges. Justify your
claim.
(a) X∞
n=1
(−1)n
√
n
(b) X∞
n=1
2
n
√
n!
(c) X∞
n=1
n!
n4 + 3
(d) X∞
n=1
1
nn
(e) X∞
n=1
(−1)n n!
2
n
3. For this question, give an example of or prove that such an example does not
exist. Prove all your claims.
(a) A convergent series P∞
n=1 an for which P∞
n=1 a
2
n
converges.
(b) A convergent series P∞
n=1 an for which P∞
n=1 a
2
n diverges.
(c) A convergent series P∞
n=1 an with an ≥ 0 for all n ∈ N, for which P∞
n=1 a
2
n
diverges.
(d) A divergent series P∞
n=1 an for which P∞
n=1 a
2
n
converges.
(e) A divergent series P∞
n=1 an for which P∞
n=1 a
2
n diverges.
4. Let (an)
∞
n=1 be a sequence such that lim inf|an| = 0. Prove that there is a
subsequence (ank
)
∞
k=1 such that P∞
k=1 ank
converges.
5. For each of the following questions, prove that for any fixed x ∈ R, the series
converges. Recall that x
0 = 1 for all x ∈ R by definition (in particular 00 = 1)
and 0! = 1. (Hint: Treat the case x = 0 and x 6= 0 separately. For x 6= 0, use
the ratio test)
(a) X∞
n=0
x
n
n!
(b) X∞
n=0
(−1)nx
2n+1
(2n + 1)!
(c) X∞
n=0
(−1)nx
2n
(2n)!
Remark 0.1. These are the definitions of exp(x),sin(x) and cos(x) respectively.

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