Use series to evaluate limx→0 e^x - (1 + x)/3x^2 limx→0 e^x - (1 + x)/

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Use series to evaluate limx→0 e^x - (1 + x)/3x^2 limx→0 e^x - (1 + x)/3x^2 = (Type a simplified fraction.)

$Use series to evaluate limx→0 e^x - (1 + x)/3x^2 limx→0 e^x - (1 + x)/3x^2 = (Type a simplified fraction.)$

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Use series to evaluate limx→0 e^x - (1 + x)/3x^2 limx→0 e^x - (1 + x)/3x^2 = (Type a simplified fraction.)

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$(a) Find the series' radius and interval of convergence. Find the values of x for which the series converges (b) absolutely and (c) conditionally ∑ ∞ n=0 x^n/√n^2 + 11 (a) The radius of convergence is (Simplify your answer.) Determine the interval of convergence. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The interval of convergence is (Type a compound inequality. Simplify your answer. Use integers or fractions for any numbers in the expression.) B. The series converges only at x = (Type an integer or a simplified fraction.) c. The series converges for all values of x. (b) For what values of x does the series converge absolutely? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The series converges absolutely for (Type a compound inequality. Simplify your answer. Use integers or fractions for any numbers in the expression.) B. The series converges absolutely at x =. (Type an integer or a simplified fraction.) c. The series converges absolutely for all values of x. (c) For what values of x does the series converge conditionally? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The series converges conditionally for (Type a compound inequality. Simplify your answer. Use integers or fractions for any numbers in the expression.) B. The series converges conditionally at x =. (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) c. There are no values of x for which the series converges conditionally.$

$(a) Find the series' radius and interval of convergence. Find the values of x for which the series converges (b) absolutely and (c) conditionally ∑ ∞ n=0 x^n/√n^2 + 11 (a) The radius of convergence is (Simplify your answer.) Determine the interval of convergence. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The interval of convergence is (Type a compound inequality. Simplify your answer. Use integers or fractions for any numbers in the expression.) B. The series converges only at x = (Type an integer or a simplified fraction.) c. The series converges for all values of x. (b) For what values of x does the series converge absolutely? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The series converges absolutely for (Type a compound inequality. Simplify your answer. Use integers or fractions for any numbers in the expression.) B. The series converges absolutely at x =. (Type an integer or a simplified fraction.) c. The series converges absolutely for all values of x. (c) For what values of x does the series converge conditionally? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The series converges conditionally for (Type a compound inequality. Simplify your answer. Use integers or fractions for any numbers in the expression.) B. The series converges conditionally at x =. (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) c. There are no values of x for which the series converges conditionally.$

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$Investigate lim x → ∞ ln⁡( x +9) ln⁡x and lim x → ∞ ln⁡( x +978) ln⁡x . Then use l'Hôpital's rule to explain what is found. What does this say about the relative rates at which the functions f(x)= ln⁡( x + a)and g(x)= ln⁡x grow? Investigate the following limits using a graphing utility. lim x → ∞ ln⁡( x +9) ln⁡x = ◻ lim x → ∞ ln⁡( x +978) ln⁡x = ◻ (Type an integer or a simplified fraction.) Use l'Hôpital's rule to evaluate the given limits. lim x → ∞ ln⁡( x +9) ln⁡x = ◻ lim x → ∞ ln⁡( x +978) ln⁡x = ◻ (Type an integer or a simplified fraction.) What can be said about the relative rates at which the functions f(x)= ln⁡( x + a)and g(x)= ln⁡x grow for any positive real number a? Choose the correct answer below. A. The function f(x)= ln⁡( x + a)grows slower than g(x)= ln⁡x as x → ∞ . B. The functions f(x)= ln⁡( x + a)and g(x)= ln⁡x grow at the same rate as x → ∞ . C. The function f(x)= ln⁡( x + a)grows faster than g(x)= ln⁡x as x → ∞ .$

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$The instructions for the given integral have two parts, one for the trapezoidal rule and one for Simpson's rule. Complete the following parts. ∫ 1 -1 (3x^2 + 7)dx. Using the trapezoidal rule a. Estimate the integral with n = 4 steps and find an upper bound for |ET|. T = (Type an exact answer. Type an integer or a simplified fraction.) An upper bound for |ET| is (Round to two decimal places as needed.) b. Evaluate the integral directly and find |ET|. ∫-1 1 (3x^2 + 7)dx = (Type an exact answer. Type an integer or a simplified fraction.) |ET| = (Round to two decimal places as needed.) c. Use the formula (|ET|/( true value )) × 100 to express |ET | as a percentage of the integral's true value. % (Round to the nearest integer as needed.) II. Using Simpson's rule a. Estimate the integral with n = 4 steps and find an upper bound for |ES|. S = (Type an exact answer. Type an integer or a simplified fraction.) An upper bound for |ES| is b. Evaluate the integral directly and find |ES |. ∫-1 1 (3x^2 + 7)dx = (Type an exact answer. Type an integer or a simplified fraction.) |ES| = c. Use the formula (|ES |/( true value) ) × 100 to express |ES| as a percentage of the integral's true value. % (Round to the nearest integer as needed.)$

$The instructions for the given integral have two parts, one for the trapezoidal rule and one for Simpson's rule. Complete the following parts. ∫ 1 -1 (3x^2 + 7)dx. Using the trapezoidal rule a. Estimate the integral with n = 4 steps and find an upper bound for |ET|. T = (Type an exact answer. Type an integer or a simplified fraction.) An upper bound for |ET| is (Round to two decimal places as needed.) b. Evaluate the integral directly and find |ET|. ∫-1 1 (3x^2 + 7)dx = (Type an exact answer. Type an integer or a simplified fraction.) |ET| = (Round to two decimal places as needed.) c. Use the formula (|ET|/( true value )) × 100 to express |ET | as a percentage of the integral's true value. % (Round to the nearest integer as needed.) II. Using Simpson's rule a. Estimate the integral with n = 4 steps and find an upper bound for |ES|. S = (Type an exact answer. Type an integer or a simplified fraction.) An upper bound for |ES| is b. Evaluate the integral directly and find |ES |. ∫-1 1 (3x^2 + 7)dx = (Type an exact answer. Type an integer or a simplified fraction.) |ES| = c. Use the formula (|ES |/( true value) ) × 100 to express |ES| as a percentage of the integral's true value. % (Round to the nearest integer as needed.)$

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