Again this scrapwork is not part of the formal proof, but it is typically necessary for finding what N should be. You might be able to do the next problem without doing any scrapwork first, but don't hesitate to do scrapwork if you need it. Problem 7.1.4. Use the definition of convergence to zero to prove the following. 1 (а) lim = 0 n→∞ n2 (b) lim 1 = 0 n As the sequences get more complicated, doing scrapwork ahead of time will becomc mo ary
Again this scrapwork is not part of the formal proof, but it is typically necessary for finding what N should be. You might be able to do the next problem without doing any scrapwork first, but don't hesitate to do scrapwork if you need it. Problem 7.1.4. Use the definition of convergence to zero to prove the following. 1 (а) lim = 0 n→∞ n2 (b) lim 1 = 0 n As the sequences get more complicated, doing scrapwork ahead of time will becomc mo ary
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help me solve problem 7.1.4
Transcribed Image Text:11:06 O
needed to use the inequality sin n| < 1.
Again this scrapwork is not part of the
formal proof, but it is typically necessary
for finding what N should be. You might
be able to do the next problem without
doing any scrapwork first, but don't
hesitate to do scrapwork if you need it.
Problem 7.1.4. Use the definition of
convergence to zero to prove the
following.
(a) lim
n→∞ n2
1
(b) lim
As the sequences get more complicated,
doing scrapwork ahead of time will
become more necessary.
Example 7.1.5. Use the definition of
convergence to zero to prove
п+4
lim
n→0 n2 +1
0.
SCRAPWORK
Given an ɛ > 0, we need to see how large
II
Transcribed Image Text:11:06
make n > . This leads to the following
definition.
Definition 7.1.1. Let
(sn) = (81, 82, s3, ...) be a sequence of
real numbers. We say that (sn) converges
to 0 and write lim,-> 8n = 0 provided for
any e > 0, there is a real number N such
that if n > N, then sn| < e.
Notes on Definition 7.1.1:
Definition 7.1.1. Let
(sn) = (81, 82, s3,...) be a sequence of
real numbers. We say that (sn)
converges to 0 and write limn→0∞ $n
provided for any ɛ > 0, there is a real
number N such that if n > N, then
Sn| < e.
in-context
1. This definition is the formal version
of the idea we just talked about; that
is, given an arbitrary distance ɛ, we
must be able to find a specific
number N such that s, is within e of
0, whenever n > N. The N is the
answer to the question of how large
is "large enough" to put s, this close
to 0.
II
V
II
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