Question 1

Find the sum of the series  .

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Question 2

Find the Maclaurin series for the function F(x) =  dt.

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Question 3

Calculate the Maclaurin series for the function F(x) =  dt and calculate F(1) correct to three decimal places.

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Question 4

Calculate the Maclaurin series for the function G(x) =  and calculate G(1) correct to 5 decimal places.

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Question 5

Given f(x) = ln(3x + 1)
(a) Find Taylor's polynomial of degree n = 2 for f about c = 0
(b) Use part (a) to estimate the value of ln(1.3).
(c) Estimate the maximum absolute error involved in the approximation obtained in part (b) above.

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(a) Find...

Question 6

How many nonzero terms of the known Maclaurin series for ln(1 + x) are needed to approximate ln(1.1) correct to 4 decimal places?

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The Maclaurin series for ln(1 + x) is x - x^2/2 + x^3/3 - x^4/4 + ... . To approximate ln(1.1) to 4 decimal places, we substitute x = 0.1 and look for the point at which additional terms contribute less than 0.00005 (half of 10^-4, due to rounding) to the sum. The third term, 0.1^3/3, is less than 0.00005, making the first two terms sufficient for the approximation. However, to ensure the approximation is correct to 4 decimal places, we include the third term, which affects the fourth decimal place due to rounding.

Question 7

How many nonzero terms of the Maclaurin series for  (x) are needed to approximate  (0.2) correct to 5 decimal places?

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Question 8

Assuming that no calculator is handy, what known Maclaurin series would you use to approximate sin(95°) using as few terms as possible, and at what point would you evaluate that series?

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Question 9

Use appropriate Maclaurin series to evaluate .

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Question 10

Use known Maclaurin series to evaluate .

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Question 11

Use known Maclaurin series to evaluate .

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Question 12

The error E in the approximation e $\approx$ satisfies  <  .

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Question 13

The error E in the approximation  $\approx$ satisfies  <  .

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Question 14

How many terms of the Maclaurin series for  are needed to approximate  with error less than 0.00005 in size?

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Question 15

For what range of x can cos x be replaced by the first three nonzero terms of its Maclaurin series to ensure the error is less than 0.00005?

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Question 16

Find the Maclaurin series (binomial series) for  .

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Question 17

Find the Maclaurin series (binomial series) for  .

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Question 18

Find the Maclaurin series (binomial series) for  (x).

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Question 19

(x) = x +  for -1 < x < 1

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Question 20

Find the Fourier series of the function f having period 2$\pi$  and defined by f(x) = 0 if -$\pi$ < x $\le$ 0 and f(x) = 1 if 0 < x   $\pi$.

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Quiz 10: Sequences, Series, and Power Series

Find the sum of the series .

Find the Maclaurin series for the function F(x) = dt.

Given f(x) = ln(3x + 1) (a) Find Taylor's polynomial of degree n = 2 for f about c = 0 (b) Use part (a) to estimate the value of ln(1.3). (c) Estimate the maximum absolute error involved in the approximation obtained in part (b) above.

How many nonzero terms of the known Maclaurin series for ln(1 + x) are needed to approximate ln(1.1) correct to 4 decimal places?

How many nonzero terms of the Maclaurin series for (x) are needed to approximate (0.2) correct to 5 decimal places?

Assuming that no calculator is handy, what known Maclaurin series would you use to approximate sin(95°) using as few terms as possible, and at what point would you evaluate that series?

Use appropriate Maclaurin series to evaluate . $Use appropriate Maclaurin series to evaluate . A) -1 B) \infty C) 3 D) 1 E) - \infty$ $Use appropriate Maclaurin series to evaluate . A) -1 B) \infty C) 3 D) 1 E) - \infty$

Use known Maclaurin series to evaluate .

Use known Maclaurin series to evaluate .

The error E in the approximation e $\approx$ $The error E in the approximation e \approx satisfies < .$ satisfies $The error E in the approximation e \approx satisfies < .$ < $The error E in the approximation e \approx satisfies < .$ .

The error E in the approximation $The error E in the approximation \approx satisfies < .$ $\approx$ $The error E in the approximation \approx satisfies < .$ satisfies $The error E in the approximation \approx satisfies < .$ < $The error E in the approximation \approx satisfies < .$ .

How many terms of the Maclaurin series for are needed to approximate with error less than 0.00005 in size?

For what range of x can cos x be replaced by the first three nonzero terms of its Maclaurin series to ensure the error is less than 0.00005?

Find the Maclaurin series (binomial series) for .

Find the Maclaurin series (binomial series) for .

Find the Maclaurin series (binomial series) for (x) .

(x) = x + for -1 < x < 1

Find the Fourier series of the function f having period 2 $\pi$ and defined by f(x) = 0 if - $\pi$ < x $\le$ 0 and f(x) = 1 if 0 < x $\pi$ .

Quiz 10: Sequences, Series, and Power Series

Find the sum of the series .

Find the Maclaurin series for the function F(x) = dt.

Calculate the Maclaurin series for the function F(x) = dt and calculate F(1) correct to three decimal places.

Calculate the Maclaurin series for the function G(x) = and calculate G(1) correct to 5 decimal places.

Given f(x) = ln(3x + 1) (a) Find Taylor's polynomial of degree n = 2 for f about c = 0 (b) Use part (a) to estimate the value of ln(1.3). (c) Estimate the maximum absolute error involved in the approximation obtained in part (b) above.

How many nonzero terms of the known Maclaurin series for ln(1 + x) are needed to approximate ln(1.1) correct to 4 decimal places?

How many nonzero terms of the Maclaurin series for (x) are needed to approximate (0.2) correct to 5 decimal places?

Assuming that no calculator is handy, what known Maclaurin series would you use to approximate sin(95°) using as few terms as possible, and at what point would you evaluate that series?

Use appropriate Maclaurin series to evaluate .

Use known Maclaurin series to evaluate .

Use known Maclaurin series to evaluate .

The error E in the approximation e ≈\approx≈ satisfies < .

The error E in the approximation ≈\approx≈ satisfies < .

How many terms of the Maclaurin series for are needed to approximate with error less than 0.00005 in size?

For what range of x can cos x be replaced by the first three nonzero terms of its Maclaurin series to ensure the error is less than 0.00005?

Find the Maclaurin series (binomial series) for .

Find the Maclaurin series (binomial series) for .

Find the Maclaurin series (binomial series) for (x) .

(x) = x + for -1 < x < 1

Find the Fourier series of the function f having period 2 π\piπ and defined by f(x) = 0 if - π\piπ < x ≤\le≤ 0 and f(x) = 1 if 0 < x π\piπ .

Use appropriate Maclaurin series to evaluate . $Use appropriate Maclaurin series to evaluate . A) -1 B) \infty C) 3 D) 1 E) - \infty$ $Use appropriate Maclaurin series to evaluate . A) -1 B) \infty C) 3 D) 1 E) - \infty$

The error E in the approximation e $\approx$ $The error E in the approximation e \approx satisfies < .$ satisfies $The error E in the approximation e \approx satisfies < .$ < $The error E in the approximation e \approx satisfies < .$ .

The error E in the approximation $The error E in the approximation \approx satisfies < .$ $\approx$ $The error E in the approximation \approx satisfies < .$ satisfies $The error E in the approximation \approx satisfies < .$ < $The error E in the approximation \approx satisfies < .$ .

Find the Fourier series of the function f having period 2 $\pi$ and defined by f(x) = 0 if - $\pi$ < x $\le$ 0 and f(x) = 1 if 0 < x $\pi$ .