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- Appendix c: analysis topics
- Convergence of sequences
This module will present an introduction into convergence and focus on what a sequence is and how it behaves as it approaches infinity.
Introduction
Insert paragraph text here.
Sequences
sequence
- A sequence is a function
defined on the positive integers
'
'. We often denote a
sequence by
Convergence of real sequences
limit
- A sequence
converges to a limit
if for
every
there is an integer
such that
We usually denote a limit by writing
or
The above definition means that no matter how small we
make
, except for a
finite number of
's, all points of the sequence are within
distance
of
.
We are given the following convergent sequence:
Intuitively we can assume the following limit:
Let us prove this rigorously. Say that we are given a
real number
. Let us choose
, where
denotes the smallest integer larger than
. Then for
we have
Thus,
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Now let us look at the following non-convergent sequence
This sequence oscillates between 1 and -1, so it will
therefore never converge.
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Source:
OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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