Suppose
is a sequence and that the numbers
,
,
, ... are limit points. Show that 0 is also a limit point.
Suppose is a sequence and that the numbers , , , ... are limit points. Show...
a) Suppose we know that the series
is convergent, where the sequence an is nonzero. Show
that the series
is divergent by applying the appropriate test.
b) Suppose we know that the series
is convergent, where the sequence cn consists of
exclusively positive terms. Show that the series
is convergent by applying the appropriate test.
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Suppose that and are Cauchy sequences. Show that the sequence is also Cauchy. Sn We were unable to transcribe this image(Sn-tn
16. lim n-> inf
24. Find the limit of the sequence
25. lim n-> inf
. limit=
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Suppose
is some sequence of holomorphic functions, which are defined on an
open set containing the closed unit disk
.
Suppose also that
converges uniformly on the unit circle
.
Show then that
converges to a holomorphic function
on
9n We were unable to transcribe this image9n aD 9n We were unable to transcribe this imageWe were unable to transcribe this image
Suppose that
a) show that
is a context free language
b) show that
for every
is also context free
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Let
be a sequence of independent random variables with
and
. Show that
in probability,
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Please keep in mind that this
is a proof using this definition of a Limit of a sequence.
We were unable to transcribe this image3.1.3 Definition A sequence X = (z.) in R is said to converge to z E R, or z is said to be a limit of (Zn), if for every ε > 0 there exists a natural number K(e) such that for allnK(e), the terms xn satisfy n- x < e. If a sequence has a...
suppose
prove that 0 is the only eigenvalue of N
(hint: fist show 0 is an eigenvalue of N, and then show if
is any
eigenvalue then =0
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Consider a second-order linear homogeneous equation
Suppose that
are two solutions. Show that
is also a solution to the equation (plug it in and use the fact
that
and
are solutions).
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Suppose that
is nonempty and bounded above. Then
has a supremum.
Note: Show that there is a least element
such that
is an upper bound for
. if
is not a least upper bound for
, show there is at least
such that
is an upper bound for
. Proceed in this way to find the supremum.
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