TBIL-Cal Manipulation of Power Series (PS3)

$\exp (2 x) = \sum_{n = 0}^{\infty} \frac{2 x^{n}}{n!} = 2 + 2 x + x^{2} + \frac{1}{3} x^{3} + \dots .$
$\exp (2 x) = \sum_{n = 0}^{\infty} \frac{2 x^{n + 1}}{n!} = 2 x + 2 x^{2} + x^{3} + \frac{1}{3} x^{4} + \dots .$
$\exp (2 x) = \sum_{n = 0}^{\infty} \frac{(2 x)^{n}}{n!} = 1 + 2 x + 2 x^{2} + \frac{4}{3} x^{3} + \dots .$
$\exp (2 x) = \sum_{n = 0}^{\infty} \frac{x^{n}}{(2 n)!} = 1 + \frac{x}{2} + \frac{x^{2}}{4} + \frac{x^{3}}{720} + \dots .$

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(b)

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What is the interval of convergence for

x

for this series?

$(- \infty, \infty) .$
$(- \frac{1}{2}, \frac{1}{2}) .$
$(0, \frac{1}{2}) .$
$(- \frac{1}{2}, \frac{1}{2}] .$

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Fact 9.3.7.

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If a power series

f (x) = \sum_{n = 0}^{\infty} a_{n} (x - c)^{n}

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is known, then for any polynomial

g (x)

the composition

f \circ g

has a power series given by

(f \circ g) (x) = f (g (x)) = \sum_{n = 0}^{\infty} a_{n} (g (x) - c)^{n}

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where the domain of convergence is transformed based upon the transformation given by

g (x) .

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For example, if

f (x)

has the domain of convergence

- 2 \leq x < 2,

then

f (2 x + 4)

has the domain of convergence:

- 2 \leq 2 x + 4 < 2

- 6 \leq 2 x < - 2

- 3 \leq x < - 1

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Activity 9.3.8.

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Suppose we wish to find the power series for the function

f (x) = \frac{1}{x} .

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(a)

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Which of the following represents the power series for

g (r) = \frac{1}{1 - r} ?

$g (r) = \sum_{n = 0}^{\infty} r x^{n} .$
$g (r) = \sum_{n = 0}^{\infty} (r x)^{n} .$
$g (r) = \sum_{n = 0}^{\infty} r^{n} .$
$g (r) = \sum_{r = 0}^{\infty} x^{r} .$

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(b)

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For what value of

r

\frac{1}{1 - r} = \frac{1}{x} ?

$r = x - 1 .$
$r = 1 - x .$
$r = x + 1 .$
$r = - x .$

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(c)

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Substituting

r

with this value, which of the following is a power series for

f (x) = \frac{1}{x} ?

$f (x) = \sum_{n = 0}^{\infty} (- x)^{n} .$
$f (x) = \sum_{n = 0}^{\infty} (1 - x)^{n} .$
$f (x) = \sum_{n = 0}^{\infty} (x - 1)^{n} .$
$f (x) = \sum_{n = 0}^{\infty} (1 + x)^{n} .$

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(d)

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Given that the domain of convergence for

r

f (r)

- 1 < r < 1,

what should be the domain of convergence for

x

f (x) ?

$- 1 < x < 1 .$
$- 2 < x < 0 .$
$- 2 < x < 2 .$
$0 < x < 2 .$

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Activity 9.3.9.

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Suppose we wish to find the power series for the function

f (x) = \frac{1}{3 - 2 x} .

Recall that

g (x) = \frac{1}{1 - r} = \sum_{n = 0}^{\infty} r^{n} .

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(a)

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For what value of

r

\frac{1}{1 - r} = \frac{1}{3 - 2 x} ?

$r = 2 x - 2 .$
$r = 2 - 2 x .$
$r = 2 x - 3 .$
$r = 3 - 2 x .$

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(b)

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Evaluating

r

at the previously found value, which of the following is the power series of

f (x) = \frac{1}{3 - 2 x} ?

$f (x) = \sum_{n = 0}^{\infty} (3 - 2 x)^{n} .$
$f (x) = \sum_{n = 0}^{\infty} (2 x - 3)^{n} .$
$f (x) = \sum_{n = 0}^{\infty} (2 - 2 x)^{n} .$
$f (x) = \sum_{n = 0}^{\infty} (2 x - 2)^{n} .$

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(c)

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Given that the interval of convergence for

r

- 1 < r < 1,

what is the interval of convergence for

x ?

$- 1 < x < \frac{3}{2} .$
$- \frac{1}{2} < x < 1 .$
$\frac{1}{2} < x < \frac{3}{2} .$
$- \frac{1}{2} < x < \frac{3}{2} .$

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Activity 9.3.10.

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Suppose we wish to find the power series for the function

f (x) = \frac{1}{1 + x^{2}} .

Recall that

g (x) = \frac{1}{1 - r} = \sum_{n = 0}^{\infty} r^{n} .

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(a)

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For what value of

r

\frac{1}{1 - r} = \frac{1}{1 + x^{2}} ?

$r = x^{2} .$
$r = - x^{2} .$
$r = 1 - x^{2} .$
$r = x^{2} - 1 .$

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(b)

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Evaluating

r

at the previously found value, which of the following is the power series of

f (x) = \frac{1}{1 + x^{2}} ?

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

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(c)

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Given that the interval of convergence for

r

- 1 < r < 1,

what is the interval of convergence for

x ?

$- 1 < x < 1 .$
$- 1 < x < 0 .$
$0 < x < 1 .$
$0 < x < 4 .$

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(d)

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How can the power series for

\frac{1}{1 + x^{2}}

be manipulated to obtain a power series for

\arctan (x) ?

Differentiate each term.
Integrate each term.
Replace $x$ with $x^{2}$ in each term.
Replace $x$ with $1 / x$ in each term.

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(e)

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Which of these power series is the result of this manipulation?

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

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Activity 9.3.11.

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What function

f (x)

has power series

f (x) = \sum_{n = 0}^{\infty} \frac{(- 1)^{n} x^{n}}{n!} = 1 - x + \frac{x^{2}}{2} - \frac{x^{3}}{6} + \dots ?

$f (x) = (- 1)^{n} e^{x} .$
$f (x) = - e^{x} .$
$f (x) = e^{- x} .$
$f (x) = - e^{- x} .$

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Activity 9.3.12.

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What function

f (x)

has power series

f (x) = \sum_{n = 0}^{\infty} \frac{x^{n + 3}}{n!} = x^{3} + x^{4} + \frac{x^{5}}{2} + \frac{x^{6}}{6} + \dots ?

$f (x) = e^{x + 3} .$
$f (x) = e^{x^{3}} .$
$f (x) = e^{3 x} .$
$f (x) = x^{3} e^{x} .$

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Fact 9.3.13.

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If a power series

f (x) = \sum_{n = 0}^{\infty} a_{n} (x - c)^{n}

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is known, then for any polynomial

g (x)

the product

f g

has a power series given by

(f g) (x) = f (x) g (x) = \sum_{n = 0}^{\infty} a_{n} g (x) (x - c)^{n}

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where the domain of convergence is the same as

f (x) .

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Activity 9.3.14.

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What function

f (x)

has power series

f (x) = \sum_{n = 3}^{\infty} x^{n} = x^{3} + x^{4} + \dots ?

$f (x) = \frac{1}{1 - 3 x} .$
$f (x) = \frac{3}{1 - x} .$
$f (x) = \frac{1}{1 - x} - x^{2} - x - 1 .$
$f (x) = \frac{x^{3}}{1 - x} .$

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Activity 9.3.15.

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The function

n (x) = e^{- x^{2}}

is one whose integrals are very important for statistics. However, it does not admit an elementary antiderivative.

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(a)

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Which of the following best represents the power series for

n (x) = e^{- x^{2}} ?

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

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(b)

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Which of the following best represents a degree 10 polynomial that approximates

n (x) ?

$n_{10} (x) = - x^{2} - x^{3} - \frac{1}{2} x^{4} - \frac{1}{6} x^{5} - \frac{1}{24} x^{6} - \frac{1}{120} x^{7} - \frac{1}{720} x^{8} - \frac{1}{5040} x^{9} - \frac{1}{40320} x^{10} .$
$n_{10} (x) = x^{2} - x^{3} + \frac{1}{2} x^{4} - \frac{1}{6} x^{5} + \frac{1}{24} x^{6} - \frac{1}{120} x^{7} + \frac{1}{720} x^{8} - \frac{1}{5040} x^{9} + \frac{1}{40320} x^{10} .$
$n_{10} (x) = 1 - x^{2} + \frac{1}{2} x^{4} - \frac{1}{6} x^{6} + \frac{1}{24} x^{8} - \frac{1}{120} x^{10} .$