Show that a bounded and monotone sequence converges. Here a
sequence
is called monotone, if it is either monotone increasing, that is
for all
or monotone decreasing, in which case
for all
.
Show that a bounded and monotone sequence converges. Here a sequence is called monotone, if it...
Suppose that
is a bounded function with following Lower and Upper
Integrals:
and
a) Prove that for every
, there exists a partition
of
such that the difference between the upper and lower sums
satisfies
.
b) Furthermore, does there have to be a subdivision such that
. Either prove it or find a counterexample and show to the
contrary.
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Prove the ratio test . What does this tell you if
exists?
(Ratio test) If
for all sufficiently large n and some
r < 1, then
converges absolutely; while if
for
all sufficiently large n, then
diverges.
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Let X be a banach space such that X= C([a,b]) where - ab+ with the sup
norm. Let x and f X. Show
that the non linear integral equation
u(x) = (sin
u(y) dy + f(x) ) has a solution u X. (the integral is
from a to b).
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sin 0, cos 0
Name the quadrant in which the angle lies
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Write down: 1) a sequence not monotone but converges to 0 2) Sequence that is bounded but not convergent
Let f(x)=
if
,
if
if
a) What is the fomain of f(x)? Write in interval notation.
b) Determine the y-intercept of the function, if any. Make sure
to justify your answer.
c) Determine the x-intercepts of the function, if any. Justify
your answer.
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4. Show that the sequence defined by a=2 An+1- 3-an satisfies () < an < 2 and is decreasing. Deduce that the sequence is convergent and find its limit.
Exercises 4.2 ove that the sequence (1 + z/n)"; n = 1, 2, 3,..., converges uni- ly in Iz <R < , for every R. What is the limit? 1, afdos se converge? diverge?
Problem 2 Show that if the sequence of numbers (an)n-1 satisfies Inlan) < oo, then the series In ancos(nx) converges uniformly on [0, 27). This means, the partial sums Sn(x) = ) ancos(nx) define a sequence of functions {sn} = that converges uniformly on [0, 271]. Hint: First show that the sequence is Cauchy with respect to || · ||00.
Given that the sequence defined by - 1 2+1 = 5-1 an is increasing and an < 5 for all n. Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)