Sequences: Limit Laws

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Name	Status	Author	Last Modified
Sequences: Limit Laws	Ready to use	Ben Brawn	04/10/2017 02:46
Sequences: Limit Laws	draft	Xiaodan Leng	11/07/2019 01:28

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Question statement

The following questions deal with the limit laws of sequences.

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			Name	Type	Generated Value

A	integer	10
B	integer	-2
C	integer	9
D	integer	-6
L	integer	-4
f	number	11
g	number	12
list	list	List of 24 items
n0	integer	20
h	integer	-7
j	number	7.5

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list[0]

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Generated value: `integer`

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list

This variable doesn't seem to be used anywhere.

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Part b) Gap-fill
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Part c) Gap-fill
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Given that

$\displaystyle\lim_{n\rightarrow\infty}a_n=\var{A}$ ,
$\displaystyle\lim_{n\rightarrow\infty}b_n=\var{B}$ ,
$\displaystyle\lim_{n\rightarrow\infty}c_n=\var{C}$ ,
$\displaystyle\lim_{n\rightarrow\infty}d_n=\var{D}$ and
$\displaystyle\lim_{n\rightarrow\infty}l_n=\var{L}$ .

Determine the following limit:

$\displaystyle\lim_{n\rightarrow\infty} \left(\frac{a_n+\var{f}b_n-c_nd_n}{l_n+\var{g}}\right)=$ Unnamed gap

Advice

The following steps through the application of the limit laws to our question

$\displaystyle\begin{align}\lim_{n\rightarrow\infty} \left(\frac{a_n+\var{f}b_n-c_nd_n}{l_n+\var{g}}\right)&=\frac{\lim_{n\rightarrow\infty}\left(a_n+\var{f}b_n-c_nd_n\right)}{\lim_{n\rightarrow\infty}\left(l_n+\var{g}\right)}\\&=\frac{\lim_{n\rightarrow\infty}(a_n)+\lim_{n\rightarrow\infty}(\var{f}b_n)-\lim_{n\rightarrow\infty}(c_nd_n)}{\lim_{n\rightarrow\infty}(l_n)+\lim_{n\rightarrow\infty}(\var{g})}\\&=\frac{\lim_{n\rightarrow\infty}(a_n)+\var{f}\lim_{n\rightarrow\infty}(b_n)-\lim_{n\rightarrow\infty}(c_n)\lim_{n\rightarrow\infty}(d_n)}{\lim_{n\rightarrow\infty}(l_n)+\lim_{n\rightarrow\infty}(\var{g})}\\&=\frac{\var{A}+\var{f}(\var{B})-(\var{C})(\var{D})}{\var{L}+\var{g}}\\&=\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{A+f*B-C*D}/{L+g}}\end{align}$

The middle sequence, $\{y_n\}$ , is squeezed by the sequence below, $\{x_n\}$ , and the sequence above, $\{z_n\}$ , which both converge to the same limit, therefore $\{y_n\}$ must also converge to the same limit. This is known as the squeeze theorem (for sequences).

The only difference between a function converging and a sequence converging is that the sequence is only defined for integers. So if the value of $a_n$ is the same as $f(n)$ and $f(x)$ converges to $\var{j}$ , then it stands to reason that the sequence $\{a_n\}$ would also converge to $\var{j}$ .

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This question is used in the following exams:

Limits of sequences by Ben Brawn in Sequences and Series.
analysis by Daniel Atcheson in Daniel's workspace.

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Ben Brawn 7 years, 6 months ago

Ben Brawn 7 years, 6 months ago

Ben Brawn 7 years, 6 months ago

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Generated value: `integer`

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Sequences: Limit Laws

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben Brawn 7 years, 6 months ago

Ben Brawn 7 years, 6 months ago

Ben Brawn 7 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`