Math 521, Lecture 2 - Spring 2009
Advanced Calculus
Homework 7
Exercise 1 Prove that the following series Panis convergent (or prove that it is divergent) when: an=
1
√2n+1 ,an=an+bn(for which values of aand bin R?), an=1
2n+1 ,an=(−1)n
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n
,an= sin 1
n2,
an= cos 1
n2,an= (−1)nsin 1
n,an= (−1)nln(1 + 1
n)
Exercise 2 Consider the series Pan, where an=1
(n+1)(n+2) .
(1) Prove Panconverges.
(2) Compute its sum (i.e. P+∞
k=0 ak).
Exercise 3 Let Panbe a divergent series of nonnegative terms and assume that (an)converges to 0.
Prove that there exists a subsequence (akn)of (an)such that Paknconverges.
Exercise 3 : Cesaro convergence Let (xn)be a sequence of real numbers. Consider the sequence (yn)
defined by yn:= x1+x2+···+xn
n.
(1) Assume that (xn)converges to 0. Prove that (yn)converges to 0(hint: separate yn in two sums,
use the fact that (xn)is bounded and that 1
nconvergences to 0for the first sum and use the fact that
(xn)convergences to 0for the second sum).
(2) Assume that (xn)converges to a real number l. Prove that (yn)converges to l.
Exercise 4: honour exercise Do the exercise (ask me) of the textbook.
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