SUGGESTIONS FOR ANALYSIS I - SERIES 7
Exercise 24
• First of all it suffices to show that
(1)
lim inf an = lim inf ak
n→+∞
n→+∞ k≥n
since the proof of the second part of the statement uses the same arguments.
Set sn = inf k≥n ak and recall that by definition
lim inf an = inf{an |n ∈ N}.
n→+∞
Thus, we want to prove that
lim inf an = lim sn
n→+∞
n→+∞
and then we have to distinguish between two cases.
– If lim inf n→+∞ an = −∞ the sequence an is not bounded from below.
Hence, there exists a monotone decreasing subsequence akn such that
akn → −∞. So that, for each n ∈ N the set {ak |k ≥ n} contains an
infinite number of elements of that subsequence. As a consequence of
this, sn = −∞ for each n ∈ N so that identity (1) is fulfilled.
– Look at the extra note.
• By definition of lim sup it is not hard to see that for each n ∈ N we have
an ≤ sup{an |n ∈ N} and bn ≤ sup{bn |n ∈ N}
so that, if we take the sum on both sides we have
an + bn ≤ sup{an |n ∈ N} + sup{bn |n ∈ N}
for each n ∈ N. Then, if we consider the supremum we find
(2)
sup{an + bn |n ∈ N} ≤ sup{an |n ∈ N} + sup{bn |n ∈ N}
and hence, since for a sequence cn we have
lim sup cn = sup{cn |n ∈ N}
n→+∞
from (2) we conclude the thesis.
• Use the suggestions of the exercise paper.
you have to
• Try with the sequence an = 2 + (−1)n . To compute aan+1
n
distinguish between even and odd numbers to obtain an explicit formula
for aan+1
.
n
Exercise 25
• Use the suggestion of the exercise paper. In particular you have to use the
decomposition in partial fractions as you see in Series 5. Then, use the
thelescopic criterion.
• For the first series, at first prove that this is an absolut convergent series.
Then, use the equality
√
√
1
n+1− n
p
p
=
√
√
n(n + 1)( n + n + 1)
n(n + 1)
1
2
SUGGESTIONS FOR ANALYSIS I - SERIES 7
so that our series becomes
√
+∞ √
+∞ X
n+1− n X
1
1
p
√ −√
=
n
n+1
n(n + 1)
n=1
n=1
(3)
then you can use the thelescopic criterion. Since this series is abolute
convergent, the Lagrange theorem for absolute convergent series ensures
you that the number found with the thelescopic criterion is the value of
convergence of the series.
For the second series, recall the formula
n
X
1 − tn+1
tk =
1−t
k=0
k
2k
+3
to apply (3).
and try to decompose the fraction 233k+1
Exercise 26
In the literature this result is known as Riemann series theorem. Look for a
proof of this.
Exercise 27
• Try to compare tha degree of the numerator and of the denominator.
• Rationalize the terms in the parenthesis and then test the convergence with
the rooth criterion or the fraction criterion.
• Rationalize the terms of the series and then compare the degrees of the
numerator and the denominator.
• You have to distinguish between the cases 0 < a < 1, a = 1 and a > 1.
Afterwards, use the convergence criteria you studied to test the convergence
of the series.
Exercise 28
Look in the literature for the proof of the Cauchy condensation test and then
apply it to study the series of point (b).