When is a sequence within d of its limit?

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Bill Foster 6 years, 5 months ago

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Question statement

Let

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

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	20
b1	integer	-7
b	integer	-7
c	integer	9
d	integer	19

			Name	Type	Generated Value

r1	integer	3
ep1	number	0.001
tval	number	5467.024691358
N_1	integer	5468

			Name	Type	Generated Value

r2	integer	2
ep2	number	0.01
N_2	integer	545

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2..20)

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Description

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Generated value: `integer`

→ Used by:

N_1
N_2
b
tval

Parts

Part a) Gap-fill
1. Gap Number entry
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Part b) Gap-fill
1. Gap Number entry
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Gap-fill

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

Find the least integer $N_1$ such that

$\left|\simplify[std]{x_n -({a} / {c})}\right| \le 10 ^ { -\var{r1}}, \textrm{ for } n \geq N_1$

Least $N_1=$ Unnamed gap

Advice

a)

To find the least $N_1$ such that all terms from the the $N_1$ ^th are within $10^{\var{-r1}}$ of the limit we proceed as follows:

$\begin{align} \left| \simplify[std]{x_n -({a} / {c})} \right| \le 10^{ -\var{r1}} &\iff \left| \simplify[std]{({a}n+{b})/({c}n+{d})-{a}/{c}}\right| \le 10 ^ { -\var{r1}} \\ &\iff \simplify[std]{abs({b*c-a*d})/({c^2}n+{c*d})}\le 10 ^ { -\var{r1}} \end{align}$

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

Rearranging this last inequality by multiplying both sides by $(\simplify[std]{{c^2}n+{c*d}}) \times 10^{\var{r1}}$ (this is positive and so the inequality does not reverse) we get:

$\simplify[std]{{c^2}n+{c*d}} \ge \var{10^r1*abs(b*c-a*d)} \iff n \ge \frac{1}{\var{c^2}}\left(\simplify[std]{{10^r1*abs(b*c-a*d)}-{c*d}}\right) = \var{tval}$

Hence the least integer value is given by $N_1=\var{N_1}$ .

b)

Using the same method you should obtain $N_2=\var{N_2}$ .

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This question is used in the following exam:

Limits of sequences by Newcastle University Mathematics and Statistics in Content created by Newcastle University.

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Bill Foster 6 years, 5 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Gap-fill

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b)

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When is a sequence within d of its limit?

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 6 years, 5 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

Generated value: `integer`