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...

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.
CO
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