1. The sequence (ak) must be convergent if:a) (ak} is real for all k 1. b) (ak) is monotone...

Question

1. The sequence (ak) must be convergent if:
a) (ak} is real for all k > 1. b) (ak) is monotone increasing and ak > 0 for all k > 1. cl (ak) is bounded. d) (ak) is bounded above and monotone increasing. e) 1.
2. Let (ak) be a decreasing sequence of positive numbers with limit zero.
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The series X (-1)k a k=0
a) might be divergent. b) has radius of convergence r = 1. c) converges to a number L which satisfies S2n+1
(Sn = X (-1)k ak) k=0 d) is absolutely convergent. e) converges to the number L = 0.
te 3. Consider I ak where each ak is positive. Which of the following k=1 statements is always valid?
a) If ak decreases to zero, then the series converges
k+1 b) If lim a - 1 and ak.o.
1 c) If ak
d) If bc- < 1 for every k, then the series diverges. ak+1 e) If > 1 for every k, then the series diverges. ak

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Robert · Accepted Answer

1. (d) is true. by monotone convergence theorem.
2. (c) is true. Converges to a number L which satisfies S2n+1

1. The sequence (ak) must be convergent if: a) (ak} is real for all k > 1. b) (ak) is monotone increasing and ak > 0 for all k > 1. cl (ak) is bounded. d) (ak) is bounded above and monotone...

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