2. Infinite Sequences and Series

Question 1

Graphing calculator required

During cleanup cycle $n$ , a filtration system removes $a_n=g(n)$ grams of a contaminant, where

$g(x)=\dfrac{12}{(x+1)^{2.3}}$

for $x\geq1$ . The graph shows the continuous model and the values $a_n$ at integer inputs. Let

$S_N=\displaystyle\sum_{n=1}^{N}a_n$ .

Round numerical answers to three decimal places.

Part (a)(i)

[1]

Write an expression for the total amount removed during the first $12$ cycles.

Part (a)(ii)

[1]

Calculate this total.

Part (b)(i)

[1]

Verify that $g$ satisfies the conditions required for the integral test on $[1,\infty)$ .

Part (b)(ii)

[1]

Determine whether $\displaystyle\sum_{n=1}^{\infty}a_n$ converges.

Part (c)(i)

[1]

Write an expression that gives lower and upper bounds for the remainder $R_{12}=\displaystyle\sum_{n=13}^{\infty}a_n$ .

Part (c)(ii)

[2]

Calculate an interval containing the total amount of contaminant that will eventually be removed.

Part (d)(i)

[1]

Write an expression that can be used to guarantee that $R_N<0.010$ gram.

Part (d)(ii)

[1]

Determine the least integer $N$ satisfying this guarantee.

Question 2

Graphing calculator not permitted

The graph shows the terms $a_n$ of an infinite series $\displaystyle\sum_{n=1}^{\infty}a_n$ .

Which conclusion is justified by the graph?

A. The series converges because the terms are bounded.

B. The series converges because the terms oscillate.

C. The series diverges because $\displaystyle\lim_{n\to\infty}a_n=1\neq0$ .

D. The series diverges because every partial sum is negative.

[1]

Select Answer

Question 3

Graphing calculator required

Let $f(x)=e^{-x^2}$ . The graph shows $f$ , its second-degree Maclaurin polynomial $P_2$ , and its fourth-degree Maclaurin polynomial $P_4$ on $-1.45\leq x\leq1.45$ . The three curves are labelled A, B, and C.

Round numerical answers to three decimal places.

Part (a)(i)

[1]

Identify the curve representing $P_2$ .

Part (a)(ii)

[1]

Identify the curve representing $P_4$ .

Part (b)

[2]

Represent $P_4(x)$ by using the Maclaurin series for $e^u$ with an appropriate substitution.

Part (c)(i)

[1]

Calculate $P_4(0.7)$ and $f(0.7)$ .

Part (c)(ii)

[1]

Calculate the absolute error when $P_4(0.7)$ is used to approximate $f(0.7)$ .

Part (d)(i)

[1]

Write an expression whose positive solution gives the largest value $b$ such that the error is $0.010$ at $x=b$ .

Part (d)(ii)

[1]

Determine this value of $b$ .

Part (e)

[1]

Explain how the graph indicates whether $P_4(x)$ is an overestimate or an underestimate of $f(x)$ for $0<x\leq b$ .

Question 4

GRAPHING CALCULATOR NOT PERMITTED

For an infinite series, $S_n$ denotes the sum of its first $n$ terms. Suppose $S_n=7-\frac{4}{n+1}$ for every positive integer $n$ . What is the sum of the infinite series?

A. $3$

B. $4$

C. $7$

D. The series diverges.

[1]

Select Answer

Question 5

Graphing calculator not permitted

A positive sequence ${u_n}$ satisfies $u_1=1$ . Define

$r_n=\dfrac{u_{n+1}}{u_n}$ .

The graph shows the values of $r_n$ . The dashed horizontal line shows the limiting level approached by the ratios.

Part (a)(i)

[1]

Calculate $u_2$ using the value of $r_1$ shown in the graph.

Part (a)(ii)

[1]

Calculate $u_3$ using the value of $r_2$ shown in the graph.

Part (a)(iii)

[1]

Calculate $u_4$ using the value of $r_3$ shown in the graph.

Part (b)(i)

[1]

Determine $\displaystyle\lim_{n\to\infty}r_n$ from the graph.

Part (b)(ii)

[2]

Justify whether $\displaystyle\sum_{n=1}^{\infty}u_n$ converges.

A second sequence is defined by $v_n=3^nu_n$ .

Part (c)(i)

[1]

Determine $\displaystyle\lim_{n\to\infty}\left|\frac{v_{n+1}}{v_n}\right|$ .

Part (c)(ii)

[1]

Justify whether $\displaystyle\sum_{n=1}^{\infty}v_n$ converges.

Part (d)

[1]

Explain why observing $r_n<1$ for only the displayed values of $n$ would not, by itself, prove that $\displaystyle\sum u_n$ converges.

Question 6

Graphing calculator not permitted

A measuring instrument is calibrated through a sequence of signed corrections. The $n$ th correction is $c_n$ millimetres, and

$S_n=\displaystyle\sum_{k=1}^{n}c_k$

is the net displacement after $n$ corrections. Each correction has half the magnitude and the opposite sign of the preceding correction. The graph shows the partial sums $S_n$ .

Part (a)(i)

[1]

Calculate $c_4$ using the partial sums in the graph.

Part (a)(ii)

[1]

Calculate $c_5$ using the partial sums in the graph.

Part (b)(i)

[1]

Represent the instrument’s eventual net displacement as an infinite geometric series.

Part (b)(ii)

[1]

Determine the exact sum of this series.

Part (c)

[2]

Explain why the odd partial sums overestimate the eventual displacement while the even partial sums underestimate it.

Part (d)(i)

[1]

Calculate the actual error when $S_4$ is used to approximate the infinite sum.

Part (d)(ii)

[1]

Verify that this error satisfies the alternating-series error bound.

Part (e)

[1]

Interpret the infinite sum in the context of the instrument.

Question 7

Graphing calculator not permitted

A patient receives a dose of $40$ milligrams of a medication at equal time intervals. Immediately before each new dose, a constant fraction of the medication present after the preceding dose remains in the body.

Let $A_n$ be the amount present immediately after the $n$ th dose. The graph shows $A_n$ for the first several doses.

Part (a)(i)

[1]

Calculate the amount remaining from the first dose immediately before the second dose.

Part (a)(ii)

[1]

Determine the fraction of the medication that remains between consecutive doses.

Part (b)(i)

[1]

Write an expression for $A_n$ as a finite geometric sum.

Part (b)(ii)

[1]

Represent $A_n$ as a closed-form expression.

Part (c)(i)

[1]

Determine $\displaystyle\lim_{n\to\infty}A_n$ .

Part (c)(ii)

[1]

Explain how the graph supports this limiting value.

Part (d)(i)

[1]

Write an expression that determines when $A_n$ is within $0.1$ milligram of its limiting value.

Part (d)(ii)

[1]

Determine the least dose number for which this occurs.

Part (e)

[1]

Interpret the limiting value in context.

Question 8

Graphing calculator not permitted

Consider the power series

$P(x)=\displaystyle\sum_{n=0}^{\infty}\frac{(x-2)^n}{4^n(n+1)}$ .

For the coefficients $c_n=\dfrac{1}{4^n(n+1)}$ , define

$q_n=\left|\dfrac{c_{n+1}}{c_n}\right|$ .

The graph shows the sequence ${q_n}$ and its limiting level.

Part (a)(i)

[1]

Determine $\displaystyle\lim_{n\to\infty}q_n$ from the graph.

Part (a)(ii)

[1]

Calculate the radius of convergence of $P$ .

Part (b)

[2]

Verify whether the series converges at $x=6$ .

Part (c)

[2]

Verify whether the series converges at $x=-2$ .

Part (d)(i)

[1]

Represent the interval of convergence on a number line, including correct endpoint notation.

Part (d)(ii)

[1]

Write the interval of convergence using interval notation.

Part (e)

[1]

Explain what can be concluded about the radius of convergence after the series is differentiated term by term.

Question 9

Graphing calculator not permitted

The graph shows the terms of the series

$\displaystyle\sum_{n=1}^{\infty}a_n$ ,

where

$a_n=\dfrac{(-1)^{n+1}}{n}$ .

The dashed curves show the positive and negative magnitude envelopes.

Part (a)(i)

[1]

Verify that the magnitudes $|a_n|$ are positive and decreasing.

Part (a)(ii)

[1]

Verify the limiting condition required by the alternating series test.

Part (a)(iii)

[1]

Determine whether $\displaystyle\sum a_n$ converges.

Part (b)(i)

[1]

Represent the absolute-value series $\displaystyle\sum|a_n|$ as a familiar series.

Part (b)(ii)

[1]

Determine whether the absolute-value series converges.

Part (c)

[1]

Justify the classification of the original series as absolutely convergent, conditionally convergent, or divergent.

Part (d)(i)

[1]

Calculate the fourth partial sum $S_4$ .

Part (d)(ii)

[1]

Estimate an interval containing the value of the infinite sum by using the alternating-series error bound.

Part (e)

[1]

Determine whether $S_4$ is an overestimate or an underestimate of the infinite sum, and justify the conclusion.

Question 10

Graphing calculator not permitted

The graph shows the partial sums

$S_n=\displaystyle\sum_{k=1}^{n}a_k$

of a geometric series. What is the value of the fourth term, $a_4$ ?

A. $-\dfrac34$

B. $\dfrac34$

C. $\dfrac{15}{4}$

D. $\dfrac92$

[1]

Select Answer

Practice Questions

2. Infinite Sequences and Series

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

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