Question 1

Evaluate the indefinite integral as a power series. $\int \tan ^ { - 1 } \left( t ^ { 2 } \right) d t$
A) $C + \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } t ^ { 2 n + 3 } } { ( 2 n + 3 ) }$
B) $C + \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } t ^ { 4 n + 3 } } { ( 2 n + 1 ) ( 4 n + 3 ) }$
C) $C + \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } t ^ { 2 n + 2 } } { ( 2 n + 1 ) }$
D) $C + \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } t ^ { 4 n + 2 } } { ( 2 n + 1 ) ( 4 n + 3 ) }$
E) $C + \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } t ^ { 4 n + 3 } } { ( 4 n + 3 ) }$

Accepted Answer

The power series expansion for $\tan^{-1}(x)$ is $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1}$. Substituting $x = t^2$, we get $\tan^{-1}(t^2) = \sum_{n=0}^{\infty} \frac{(-1)^n t^{4n+2}}{2n+1}$. Integrating term by term gives $\int \tan^{-1}(t^2) dt = C + \sum_{n=0}^{\infty} \frac{(-1)^n t^{4n+3}}{(2n+1)(4n+3)}$.

Question 2

Given the series $\sum _ { m = 1 } ^ { \infty } \frac { 3 m } { 4 ^ { m } ( 3 m + 5 ) }$ estimate the error in using the partial sum $S _ { 8 }$ by comparison with the series $\sum _ { m = 9 } ^ { \infty } \frac { 1 } { 4 ^ { m } }$ .
A) $R _ { 8 } \leq 2.6130051$
B) $R _ { 8 } \geq 0.0000052$
C) $R _ { 8 } \leq 0.0000051$
D) $R _ { 8 } \geq 0.0000051$
E) $$R _ { 8 } \leq 0.000005$$

Accepted Answer

The error is bounded by the sum of the remaining terms, which is less than the sum of the terms of the comparison series starting with the first term omitted.

Question 3

Find the sum of the series. $\sum _ { n = 0 } ^ { \infty } \frac { 5 ^ { n } } { 6 ^ { n } n ! }$

Accepted Answer

$e ^ { 5 / 6 }$

Question 4

Find a power series representation for $f ( t ) = \ln ( 14 - t )$
A) $\ln 14 - \sum _ { n = 1 } ^ { \infty } \frac { t ^ { n } } { 14 ^ { n } }$
B) $\ln 14 - \sum _ { n = 1 } ^ { \infty } \frac { t ^ { n } } { n 14 ^ { n } }$
C) $\sum _ { n = 0 } ^ { \infty } \frac { t ^ { n } } { n 14 ^ { n } }$
D) $\sum _ { n = 1 } ^ { \infty } \frac { 14 t ^ { n } } { n ^ { n } }$
E) $\ln 14 + \sum _ { n = 1 } ^ { \infty } \frac { t ^ { 2 n } } { 14 ^ { n } }$

Accepted Answer

We know that the power series representation of ln(1+x) is:
$$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}x^n$$
By replacing x with -t/14 and multiplying the series by -1, we get:
$$\ln(14-t)=\ln 14 + \sum_{n=1}^{\infty}\frac{t^n}{n14^n}$$
This matches choice B. Therefore, the correct answer is B.

Question 5

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. $f ( x ) = 5 e ^ { - x ^ { 2 } } \cos 4 x$
A) $5 \left( 1 - 17 x ^ { 2 } + \frac { 115 } { 6 } x ^ { 4 } \right)$
B) $5 \left( 1 - 9 x ^ { 2 } + \frac { 115 } { 6 } x ^ { 4 } \right)$
C) $5 \left( 1 - 9 x + \frac { 115 } { 6 } x ^ { 4 } \right)$
D) $5 \left( 1 - 9 x ^ { 2 } + \frac { 97 } { 6 } x ^ { 4 } \right)$
E) $5 \left( 1 - 17 x ^ { 2 } + \frac { 67 } { 6 } x ^ { 4 } \right)$

Accepted Answer

The Maclaurin series for $e^{-x^2}$ is $1 - x^2 + \frac{x^4}{2} + \cdots$ and for $\cos 4x$ is $1 - \frac{(4x)^2}{2} + \frac{(4x)^4}{24} + \cdots$. Multiplying these series and simplifying gives the first three nonzero terms: $1 - 9x^2 + \frac{115}{6}x^4$. Multiplying by 5 gives the correct answer.

Question 6

Use series to approximate the definite integral to within the indicated accuracy. $\int _ { 0 } ^ { 05 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad \mid \text { error } \mid < 0.001$
A) 0.0354
B) 0.0125
C) 0.0625
D) 0.1447
E) 0.2774

Accepted Answer

The integral can be approximated using the series expansion of $e^{-x^2}$. The series converges quickly, and calculating a few terms gives an approximation of 0.0354, which satisfies the error condition.

Question 7

Find the Taylor series for $f ( x )$ centered at the given value of 
a. Assume that f has a power series expansion. Also find the associated radius of convergence.
 $f ( x ) = x ^ { 4 } - 3 x ^ { 2 } + 1 , a = 1$

Accepted Answer

The answer of Find the Taylor series for \(f (...

Question 8

Use the power series for $f ( x ) = \sqrt [ 3 ] { 5 + x }$ to estimate $\sqrt [ 3 ] { 5.07 }$ correct to four decimal places.
A) $1.7179$
B) $1.7189$
C) $1.7195$
D) $1.7156$
E) $1.7200$

Accepted Answer

The answer of Use the power series for \(f (...

Question 9

Find the Maclaurin series for f (x) using the definition of the Maclaurin series. $f ( x ) = x \cos ( 4 x )$
A) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { 2 n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$
B) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { 2 n } x ^ { 2 n + 1 } } { n ! }$
C) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } 5 ^ { 2 n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$
D) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$
E) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { 2 n } x ^ { 2 n } } { ( 2 n ) ! }$

Accepted Answer

The Maclaurin series for f(x) is given by:$$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n$$where $f^{(n)}(0)$ is the nth derivative of f(x) evaluated at x = 0.For $f(x) = x \cos(4x)$, we have:$$f'(x) = \cos(4x) - 4x \sin(4x)$$$$f''(x) = -4\sin(4x) - 16x \cos(4x)$$$$f'''(x) = -16\cos(4x) + 64x \sin(4x)$$$$f^{(4)}(x) = 64\sin(4x) + 256x \cos(4x)$$and so on.Evaluating these derivatives at x = 0, we get:$$f(0) = 0$$$$f'(0) = 1$$$$f''(0) = -4$$$$f'''(0) = 0$$$$f^{(4)}(0) = 64$$and so on.Therefore, the Maclaurin series for $f(x) = x \cos(4x)$ is:$$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n = x - \frac{4}{2!} x^2 + \frac{64}{4!} x^4 - \cdots = \sum_{n=0}^\infty \frac{(-1)^n 5^{2n} x^{2n+1}}{(2n)!}$$

Question 10

Evaluate the indefinite integral as an infinite series. $\int \frac { 5 \left( e ^ { x } - 1 \right) } { x } d x$

Accepted Answer

The answer of Evaluate the indefinite integral as an infinite...

Question 11

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { x } { \sqrt { 9 + x ^ { 2 } } }$
A) $| x | < 1$
B) $| x | < 10$
C) $| x | < 6$
D) $| x | < 9$
E) $| x | < 3$

Accepted Answer

The function can be rewritten as $x (9 + x^2)^{-1/2}$. Using the binomial series expansion for $(1 + u)^n$ where $u = \frac{x^2}{9}$ and $n = -1/2$, the series converges when $|u| < 1$, which translates to $|x^2/9| < 1$ or $|x| < 3$.

Question 12

Find the Maclaurin series for f and its radius of convergence. $f ( x ) = 2 \ln ( 1 - x )$

Accepted Answer

The answer of Find the Maclaurin series for f and...

Question 13

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { 1 } { ( 4 + x ) ^ { 5 } }$

Accepted Answer

The answer of Use the binomial series to expand the...

Question 14

Use series to evaluate the limit correct to three decimal places. $\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$ Select the correct answer.
A) 118.933
B) 114.133
C) 34.3233
D) 114.333
E) 115.933

Accepted Answer

The series expansion for $\tan^{-1}(x)$ around $x=0$ is $x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots$. Substituting $7x$ for $x$, we get $\tan^{-1}(7x) = 7x - \frac{(7x)^3}{3} + \cdots$. The expression becomes $\frac{7x - (7x - \frac{343x^3}{3})}{x^3} = \frac{\frac{343x^3}{3}}{x^3} = \frac{343}{3} = 114.333$.

Question 15

Find a power series representation for the indefinite integral. $\int \frac { \sin 9 x } { x } d x$

Accepted Answer

The answer of Find a power series representation for the...

Question 16

Find a power series representation for the function. $f ( y ) = \ln \left( \frac { 11 + y } { 11 - y } \right)$
A) $\sum _ { n = 0 } ^ { \infty } 22 y ^ { 2 n + 1 }$
B) $\sum _ { n = 0 } ^ { \infty } \frac { 2 y ^ { 2 n + 1 } } { 11 ^ { n + 1 } ( 2 n + 1 ) }$
C) $\sum _ { n = 0 } ^ { \infty } \frac { y ^ { 2 n + 1 } } { 11 ^ { n + 1 } ( n + 1 ) }$
D) $\sum _ { n = 7 } ^ { \infty } \frac { 22 y ^ { 2 n + 1 } } { 22 }$
E) $\sum _ { n = 0 } ^ { \infty } \frac { y ^ { 2 n + 1 } } { 11 }$

Accepted Answer

The function can be expressed as a power series by using the formula for the logarithm of a quotient and recognizing it as a geometric series. The series expansion for $\ln\left(\frac{1+x}{1-x}\right)$ is $\sum_{n=0}^{\infty} \frac{2x^{2n+1}}{2n+1}$. Substituting $x = \frac{y}{11}$ gives the correct series.

Question 17

Find the Maclaurin series for $f ( x )$ using the definition of a Maclaurin serires. $f ( x ) = ( 4 + x ) ^ { - 3 }$

Accepted Answer

The answer of Find the Maclaurin series for \(f (...

Question 18

Find a power series representation for the function and determine the radius of convergence. $f ( x ) = \arctan \left( \frac { x } { 4 } \right)$

Accepted Answer

The answer of Find a power series representation for the...

Question 19

Evaluate the function $f ( x ) = \cos x$ by a Taylor polynomial of degree $4$ centered at $a = 0$ , and $x = \frac { \pi } { 4 }$ .
A) $0.7074$
B) $4.2074$
C) $3.2074$
D) $2.2074$
E) $1.2074$

Accepted Answer

The Taylor polynomial of degree 4 for $f(x) = \cos x$ centered at $a = 0$ is $1 - \frac{x^2}{2} + \frac{x^4}{24}$. Evaluating this at $x = \frac{\pi}{4}$ gives approximately $0.7074$.

Question 20

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { n } } { n ! }$
A) $R = 7 , I = ( - 7,7 )$
B) $R = 0 , I = \{ 0 \}$
C) $R = 7 , I = [ - 7,7 ]$
D) $R = \infty , I = ( - \infty , \infty )$

Accepted Answer

The given series is an exponential series of the form $e^{7x}$, which has an infinite radius of convergence and converges for all real numbers $x$.

Quiz 11: Infinite Sequences and Series

Evaluate the indefinite integral as a power series. ∫tan⁡−1(t2) dt\int \tan ^ { - 1 } \left( t ^ { 2 } \right) d t∫tan−1(t2) dt

Find the sum of the series. ∑n=0∞5n6nn!\sum _ { n = 0 } ^ { \infty } \frac { 5 ^ { n } } { 6 ^ { n } n ! }∑n=0∞​6nn!5n​

Find a power series representation for f(t) =ln⁡(14−t) f ( t ) = \ln ( 14 - t ) f(t) =ln(14−t)

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. f(x) =5e−x2cos⁡4xf ( x ) = 5 e ^ { - x ^ { 2 } } \cos 4 xf(x) =5e−x2cos4x

Use series to approximate the definite integral to within the indicated accuracy. ∫005x2e−x2dx∣ error ∣<0.001\int _ { 0 } ^ { 05 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad \mid \text { error } \mid < 0.001∫005​x2e−x2dx∣ error ∣<0.001

Find the Taylor series for f(x)f ( x )f(x) centered at the given value of a. Assume that f has a power series expansion. Also find the associated radius of convergence. f(x)=x4−3x2+1,a=1f ( x ) = x ^ { 4 } - 3 x ^ { 2 } + 1 , a = 1f(x)=x4−3x2+1,a=1

Use the power series for f(x) =5+x3f ( x ) = \sqrt [ 3 ] { 5 + x }f(x) =35+x​ to estimate 5.073\sqrt [ 3 ] { 5.07 }35.07​ correct to four decimal places.

Find the Maclaurin series for f (x) using the definition of the Maclaurin series. f(x) =xcos⁡(4x) f ( x ) = x \cos ( 4 x ) f(x) =xcos(4x)

Evaluate the indefinite integral as an infinite series. ∫5(ex−1)xdx\int \frac { 5 \left( e ^ { x } - 1 \right) } { x } d x∫x5(ex−1)​dx

Use the binomial series to expand the function as a power series. Find the radius of convergence. x9+x2\frac { x } { \sqrt { 9 + x ^ { 2 } } }9+x2​x​

Find the Maclaurin series for f and its radius of convergence. f(x)=2ln⁡(1−x)f ( x ) = 2 \ln ( 1 - x )f(x)=2ln(1−x)

Use the binomial series to expand the function as a power series. Find the radius of convergence. 1(4+x)5\frac { 1 } { ( 4 + x ) ^ { 5 } }(4+x)51​

Use series to evaluate the limit correct to three decimal places. lim⁡x→07x−tan⁡−17xx3\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }limx→0​x37x−tan−17x​ Select the correct answer.

Find a power series representation for the indefinite integral. ∫sin⁡9xxdx\int \frac { \sin 9 x } { x } d x∫xsin9x​dx

Find a power series representation for the function. f(y) =ln⁡(11+y11−y) f ( y ) = \ln \left( \frac { 11 + y } { 11 - y } \right) f(y) =ln(11−y11+y​)

Find the Maclaurin series for f(x)f ( x )f(x) using the definition of a Maclaurin serires. f(x)=(4+x)−3f ( x ) = ( 4 + x ) ^ { - 3 }f(x)=(4+x)−3

Find a power series representation for the function and determine the radius of convergence. f(x)=arctan⁡(x4)f ( x ) = \arctan \left( \frac { x } { 4 } \right)f(x)=arctan(4x​)

Evaluate the function f(x) =cos⁡xf ( x ) = \cos xf(x) =cosx by a Taylor polynomial of degree 444 centered at a=0a = 0a=0 , and x=π4x = \frac { \pi } { 4 }x=4π​ .

Find the radius of convergence and the interval of convergence of the power series. ∑n=0∞(7x) nn!\sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { n } } { n ! }∑n=0∞​n!(7x) n​

Evaluate the indefinite integral as a power series. $\int \tan ^ { - 1 } \left( t ^ { 2 } \right) d t$

Find the sum of the series. $\sum _ { n = 0 } ^ { \infty } \frac { 5 ^ { n } } { 6 ^ { n } n ! }$

Find a power series representation for $f ( t ) = \ln ( 14 - t )$

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. $f ( x ) = 5 e ^ { - x ^ { 2 } } \cos 4 x$

Use series to approximate the definite integral to within the indicated accuracy. $\int _ { 0 } ^ { 05 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad \mid \text { error } \mid < 0.001$

Find the Taylor series for $f ( x )$ centered at the given value of a. Assume that f has a power series expansion. Also find the associated radius of convergence. $f ( x ) = x ^ { 4 } - 3 x ^ { 2 } + 1 , a = 1$

Use the power series for $f ( x ) = \sqrt [ 3 ] { 5 + x }$ to estimate $\sqrt [ 3 ] { 5.07 }$ correct to four decimal places.

Find the Maclaurin series for f (x) using the definition of the Maclaurin series. $f ( x ) = x \cos ( 4 x )$

Evaluate the indefinite integral as an infinite series. $\int \frac { 5 \left( e ^ { x } - 1 \right) } { x } d x$

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { x } { \sqrt { 9 + x ^ { 2 } } }$

Find the Maclaurin series for f and its radius of convergence. $f ( x ) = 2 \ln ( 1 - x )$

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { 1 } { ( 4 + x ) ^ { 5 } }$

Use series to evaluate the limit correct to three decimal places. $\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$ Select the correct answer.

Find a power series representation for the indefinite integral. $\int \frac { \sin 9 x } { x } d x$

Find a power series representation for the function. $f ( y ) = \ln \left( \frac { 11 + y } { 11 - y } \right)$

Find the Maclaurin series for $f ( x )$ using the definition of a Maclaurin serires. $f ( x ) = ( 4 + x ) ^ { - 3 }$

Find a power series representation for the function and determine the radius of convergence. $f ( x ) = \arctan \left( \frac { x } { 4 } \right)$

Evaluate the function $f ( x ) = \cos x$ by a Taylor polynomial of degree $4$ centered at $a = 0$ , and $x = \frac { \pi } { 4 }$ .

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { n } } { n ! }$