When does a sequence get within $d$ of its limit?

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Name	Status	Author	Last Modified
When does a sequence get within $d$ of its limit?	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:51
Tom's copy of When does a sequence get within d of its limit?	draft	Tom Stallard	20/06/2016 18:30
James's copy of When does a sequence get within $d$ of its limit?	draft	James Denholm-Price	31/03/2019 16:47

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Question statement

Let

$x_n=\simplify[std]{({a}n^2+{b})/({c}n^2+{d})}, \quad n=1,2,3, \ldots$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	15
c	number	14
b	integer	-2
s1	integer	-1
b1	integer	-2
d	number	5
r	integer	6
n	integer	725
tval	number	724.9205797457
mono	integer	1
t1	string	Increasing
t2	string	Decreasing
t3	string	Neither
u1	string	Increasing
u2	string	Decreasing
v1	string	increasing
v2	string	decreasing
w1	string	increasing

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Part a) Gap-fill
1. Gap Mathematical expression
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Part b) Gap-fill
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Part c) Choose one from a list

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Ask the student a question, and give any hints about how they should answer this part.

Find the limit $\ell$ of $\{x_n\}$ . Input as a fraction or an integer.

Limit $\ell=$ Unnamed gap

Advice

a)

To begin with, the limit $\ell$ is obtained by dividing top and bottom by $n^2$ :

$\simplify[std]{({a}n^2+{b})/({c}n^2+{d})-{a}/{c}} = \simplify[std]{({a}+{b}/n^2) /({c}+{d}/n^2)} \to \simplify[std]{({a})/({c})}$ as

$n \to \infty$ , so

$\displaystyle \ell= \simplify[std]{{a}/{c}}$ .

b)

To find the least $N$ such that all terms from the $N$ th are less than $10^{\var{-r}}$ from the limit we proceed as follows:

$\begin{align} \left|\simplify[std]{x_n -({a} / {c})}\right| < 10 ^ { -\var{r}} &\iff \left|\simplify[std]{({a}n^2+{b})/({c}n^2+{d})-{a}/{c}}\right| < 10 ^ { -\var{r}} \\ &\iff \simplify[std]{abs({b*c-a*d})/({c^2}n^2+{c*d})} <10 ^ { -\var{r}} \end{align}$

(We can get rid of the absolute value in the denominator as $\simplify[std]{{c^2}n^2+{c*d}} \gt 0$ , $\forall n=1,2,3,\ldots$ )

Rearranging this last inequality by multiplying both sides by $(\simplify[std]{{c^2}n^2+{c*d}})10^{\var{r}}$ (this is positive and so the inequality does not reverse), we get:

$\simplify[std]{{c^2}n^2+{c*d}} > \var{10^r*abs(b*c-a*d)} \iff n^2 > \frac{1}{\var{c^2}}\left(\simplify[std]{{10^r*abs(b*c-a*d)}-{c*d}}\right)=\var{tval^2} \iff n> \var{tval}$

Hence the least integer value is given by $N=\var{N}$ .

c)

Given $x_n = \dfrac{an^2+b}{cn^2+d}, c \gt 0, d\gt 0$ it can be shown that $x_n \leq x_{n+1} \iff \dfrac{b}{d} \leq \dfrac{a}{c}$ . Here $\dfrac{b}{d}=\dfrac{\var{b}}{\var{d}}$ and $\dfrac{a}{c}=\dfrac{\var{a}}{\var{c}}$ . Therefore the sequence will be increasing if $\dfrac{\var{b}}{\var{d}} \leq \dfrac{\var{a}}{\var{c}}$ and decreasing if $\dfrac{\var{b}}{\var{d}} \geq \dfrac{\var{a}}{\var{c}}$ . Hence the sequence is $\var{w1}$ .

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When does a sequence get within $d$ of its limit?

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Bill Foster 6 years, 6 months ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Generated value: `integer`

Parts

Gap-fill

a)

b)

To find the least $N$ such that all terms from the $N$ th are less than $10^{\var{-r}}$ from the limit we proceed as follows:

c)

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When does a sequence get within dd of its limit?

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

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Comment

Checkpoint description

Bill Foster 6 years, 6 months ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

To find the least NN such that all terms from the NNth are less than 10{−r}10^{\var{-r}} from the limit we proceed as follows:

c)

When does a sequence get within $d$ of its limit?

Generated value: `integer`

To find the least $N$ such that all terms from the $N$ th are less than $10^{\var{-r}}$ from the limit we proceed as follows: