12. Use the Maclaurin expansion for e-t to express the function F(2) = dt as an...
3) Let F(x) = {* In In(1+t) dt. t (a) Find the Maclaurin series for F: (b) Use the series in part (a) to evaluate F(-1) exactly and use the result to state its interval of convergence. (c) Approximate F(1) to three decimals. (Hint: Look for an alternating series. )
7. (a) Use the well known Maclaurin series expansion for the cosine function: f (x ) = cos x = 1 x? 2! + 4! х 6! + (-1)" (2n)! . * 8! 0 and a substitution to obtain the Maclaurin series expansion for g(x) = cos (x²). Express your formula using sigma notation. (b) Use the Term-by-Term Integration Theorem to obtain an infinite series which converges to: cos(x) dx . y = cos(x²) (c) Use the remainder theorem associated...
10. 1.92 points RogacalcE13 10 7.047 y Notes Ask Your leacher Consider the following function. (a) Use the Maclaurin expansion for et to express the function Fox) as an alternating power series in x. F(x) (b) Find an N such that the partial sum s of the Maclaurin series approximates the integral for x1.1 to within an error of at most 0.001 (c) Compute sw using the N found above. (Round your answer to six decimal places.) SN 7. 01.92...
In the following, we will tse a kmown power series to approximate 1/2 arctan(r) dr to within 0.00001 of the actual value of the definite integral (a) [2pt] Use a known power series representation to express (ctan(x) as a Maclaurin series. What is the radius of the series convergence? 1/2 (b) [4pts] Use your answer from part (a) to express(r) dr as an alternating series (c) [6pts] Your series in part (b) will converge by the Alternating Series Test. (You...
10. (4 points) a. Use the Maclaurin series for function to find the first three nonzero terms of the Maclaurin series for the following function. b. Then use these terms (above) to approximate the integral 10.(4 points) a. Use the Maclaurin series for function f(x) = tan-?x to find the first three nonzero terms of the Maclaurin series for the following function g(x) = x tan-1x3. b. Then use these terms (above) to approximate the integral $0.5 x tan-1 x...
find the maclaurin series for the given function f.) = cos(t)dt Jo
Use a Maclaurin series in this table to obtain the Maclaurin series for the given function. f(x) = 7x cos(2x2) (c) Use part (b) to find a power series for AUX) - 1621) 1x) - -1) ( 2.6 +1 +3 What is the radius of convergence, R? R-6 Find the Maclourin series for FUX) using the definition of a Maclaurin series. Assume that f has a power series expansion. Do not show that Ra(x) +0.1 Rox) = sin( Find the...
Question 2, non-calculator Question 1, calculator The curve C in the x-y-plane is given parametrically by (x(t), y(t), where dr = t sine) and dv = cos| t The Maclaruin series for a function f is given by r" for 1 sts 6 a) Use the ratio test to find the interval of convergence of the Maclaurin series for f a) Find the slope of the line tangent to the curve C at the point where t 3. b) Let...
Please answer all, be explanatory but concise. Thanks. Consider the function f(x) = e x a. Differentiate the Taylor series about 0 of f(x). b. ldentify the function represented by the differentiated series c. Give the interval of convergence of the power series for the derivative. Consider the differential equation y'(t) - 4y(t)- 8, y(0)4. a. Find a power series for the solution of the differential equation b. ldentify the function represented by the power series. Use a series to...
(2) Convert dix bo0 series using power seres expansion Operations 1+x² Convert 1 dx to a series using power series expansion +2x (PART TWO) operahus. (Expres.your final answer in Einstein) U the equality i-cosex) = 1-(1-32X+***** xº+coo) simplifier to the power sorres expansion . 10.8.a e thes, or equivalently, (.1-cos(x)) = 20 Now let s donute the integral - ws(x) ok allsing power series.... 2r la comert s to an alternating enes with ratsonal terms an satisfying the AST andition...