184 Chapter4
(c) lim (Vn^2 +n - n) =^1 / 2
n->oo
- Prove that if Arg Zn -7 e and lzn I -7 r' then Zn -7 re ;e.
6. Show that the sequence defined by Zn= inlnn (n = 1,2,3, ... ) 1s
'divergent.
- Evaluate limn->oo n!/3n.
- A point c is said to be an accumulation point of a sequence {Zn} if there
are infinitely many terms of the sequence in any given neighborhood
N.( c) ( e > 0). Find the set of accumulation points of each of the
following sequences.
(a) {in}
(c) {in+^1 sin n
2
7r}
( l)nn
(e) Zn= 2n + 1
(g) Zn= .!. + (-lti
n
(b) {cosn7r + _n_i}
n+l
1
( d) Zn = 1 + ( -1 r + -
n
1
(f) z - -----
n-l+(-l)n+l/n
(h) Zn = enrri/4
- In parts (d), (e), and (f) of problem 8, find limsupzn, liminfzn,
lub{zn}, and glb{zn}.
10. If a sequence {zn} has an accumulation point c, show that there is a
subsequence { Znk} that converges to c.
- Prove that if the range of {zn} is finite, {zn} has a convergent
subsequence.
- Prove that if the range of {zn} is bounded, {zn} has at least a finite
accumulation point.
- Let A be an infinite set of complex numbers. Prove that b is an accu-
mulation point of A iff there exists a sequence {Zn} of distinct points
in A such that limn->oo Zn = b.
- Give an example of a sequence that does not converge but has just one
accumulation point in C.
- Use Definition 4.7 to show that {(-lr /n!} is a Cauchy sequence.
16. Consider two se.quences {an} and {bn}· If for all pairs of indices the
inequality lbm - bnl :S lam -anl holds, the sequence {bn} is called
a contraction of the sequence {an} ([1], p. 35). Show that if {an}
converges, so does {bn}·
- Show that if { x n} is bounded above and x n :S x n+ 1 for all n, then
limn->oo Xn exists.
- Find a necessary and sufficient condition relating an and bn so that
the sequence {Aan + Bbn} be convergent, A and B being nonzero
constants.
B. Volk, [Amer. Math. Monthly, 74 (1967), 333-335]
- Use Stolz's rule to show the following.
