Question: Use the power series for tan-1x
Use the power series for tan-1x to prove the following expression for π as the sum of an infinite series:
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> Find the Maclaurin series for f (x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn (x) → 0.] Also find the associated radius of convergence. f(x): = sin 7x
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> Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x) = sin'x [Hint: Use sin'x = }(1 – s 2x).] cos %3D
> Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x) = x² In(1 + x³)
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> If f(n)(0) = (n + 1)! For n = 0, 1, 2, … find the Maclaurin series for f and its radius of convergence.
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> Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x) = cos(Tx/2)
> Use the binomial series to expand the function as a power series. State the radius of convergence. 1 (2 + x) 13
> Prove that the series obtained in Exercise 16 represents sin x for all x. Exercise 16: Find the Taylor series for f (x) centered at the given value of f. [Assume that has a power series expansion. Do not show that Rn (x) →0.] f(x
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> Find the area of the region that is bounded by the given curve and lies in the specified sector. r = sin 0, 7/3 < 0 < 2m/3
> Find the Taylor series for f (x) centered at the given value of f. [Assume that has a power series expansion. Do not show that Rn (x) →0.] f(x) 3 сos x, а a = T
> Find the Taylor series for f (x) centered at the given value of f. [Assume that has a power series expansion. Do not show that Rn (x) →0.] f(x) = 1/x, a = -3
> Find the Taylor series for f (x) centered at the given value of f. [Assume that has a power series expansion. Do not show that Rn (x) →0.]
> Find the Taylor series for f (x) centered at the given value of f. [Assume that has a power series expansion. Do not show that Rn (x) →0.] f(x) = x – x', a= -2
> Find the Taylor series for f (x) centered at the given value of f. [Assume that has a power series expansion. Do not show that Rn (x) →0.] f(x) = x* – 3x? + 1, a = 1
> If f (x) = ∑∞n=0 bn (x – 5)n for all x, write a formula for bs.
> Find a power series representation for the function and determine the interval of convergence. 1 + x f(x) 1- x
> Find a power series representation for the function and determine the interval of convergence. f(x) = 2x? + 1
> Find a power series representation for the function and determine the interval of convergence. f(x) = 9 + x?
> Find a power series representation for the function and determine the interval of convergence. f(x) = x + 10
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> Find a power series representation for the function and determine the interval of convergence.
> (a). By completing the square, show that (b). By factoring x3 + 1 as a sum of cubes, rewrite the integral in part (a). Then express 1/ (x3 + 1) as the sum of a power series and use it to prove the following formula for π: dx C1/2 Jo x -
> Find a power series representation for the function and determine the interval of convergence. 3 f(x) 1- x*
> (a). Starting with the geometric series∑∞n=0xn, find the sum of the series (b). Find the sum of each of the following series. (c). Find the sum of each of the following series. 2 nx"-I -1 |x| (i) E nx", |x|<1
> The Bessel function of order 1 is defined by (a). Show that j1 satisfies the differential equation (b). Show that j10 (x) = -j1(x). Ji(x) = E (-1)^x²«+1 n!(n + 1)!22n+1 A-0 x³J*(x) + xJ{(x) + (x² – 1)J1(x) = 0
> (a). Show that j0 (the Bessel function of order 0 given in Example 4) satisfies the differential equation (b). Evaluate f10 j0 (x) dx correct to three decimal places. X²J®(x) + xJ¿(x) + x²J(x) = 0
> Show that the function is a solution of the differential equation f"(x) + f (x) = 0 (-1)"x2" f(x) = E %3! A-0 (2n)!
> Use the result of Example 7 to compute arctan 0.2 correct to five decimal places.
> Use a power series to approximate the definite integral to six decimal places. .2 r0.3 dx 1 + x 4 Jo
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> Use a power series to approximate the definite integral to six decimal places. 1 dx Jo 1+ x* r0.2
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> Evaluate the indefinite integral as a power series. What is the radius of convergence? X - tan х — -dx .3
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> Evaluate the indefinite integral as a power series. What is the radius of convergence? t dt 1- 18
> Find a power series representation for f, and graph f and several partial sums sn (x) on the same screen. What happens as n increases? f(x) = tan-(2x)
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> Find a power series representation for the function and determine the radius of convergence. 1 + x f(x) = (1 – x)?
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> Use the following steps to prove (17). (a). Let g (x) = ∑∞n=0 (k/n) xn. Differentiate this series to show that (b). Let h (x) = (1 + x)-k g (x) and show that h'(x) = 0. (c). Deduce that g (x) = (1 + x)k. kg(x) 1
> (a). Use Equation 1 to find a power series representation for f (x) = ln (1 – x). What is the radius of convergence? (b). Use part (a) to find a power series for f (x) = x ln (1 – x). (c). By putting x = 1/2 in your result from part (a), express ln 2 as
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> Prove Taylor’s Inequality for n = 2, that is, prove that if |f"'(x)| M | R:(x) | <x - a for |x – a|<d 6.
> Find the sum of the series. 1/ 1∙2 – 1/3∙23 + 1/5∙ 25 – 1/7∙ 27 +…
> Find the sum of the series. 3 + 9/2! + 27/3! + 81/4! +…
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> Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function. y = e* In(1 + x)
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> Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function. y sin x
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> Use series to evaluate the limit. limx→0 1 – cos x/1 + x - ex
> Use series to evaluate the limit. limx→0 x – 1 ln (1 + x)/x2
> Use series to approximate the definite integral to within the indicated accuracy. x²e* dx ([error|< 0.001) Jo
> Use series to approximate the definite integral to within the indicated accuracy. L" VI + x* dx (lerror| < 5 x 10-9) *0.4
> Use series to approximate the definite integral to within the indicated accuracy. *0.2 " (tan-(x) + sin(x³)] dx (five decimal places)
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