Suppose G(r) is the generating function of a sequence (an)o, satisfying G'(r) =rG(r), G(0) = 1. i. Use the ODE to show that G'(0) = 0 and use this information, together with G(0) = 1, to find ao and a₁.
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- Let f(x) = (x − 3)^5 and x0 is not equal to 3. For each n ≥ 0, determine xn+1 from xn by using Newton’s method for finding the root of the equation f(x) = 0. Show that the sequence {xn} converges to 3 linearly with rate 4/5.Solve until the 6th derivative and provide the SUMMATION NOTATION of the taylor series as the final answer. f(x) = ln 4x ; c = ¼Determine the coefficients of the x3 term of the Taylor series about the point 0 of the function f(x) = 1/(3 -x)
- Express the function as the sum of a power series by first using partial fractions. f(x) = 8 x2 − 2x − 15Find the power series representation for the derivative of the functionSuppose that ∞ n = 0 anxn converges to a function y such that y'' − 2y' + y = 0 where y(0) = 0 and y'(0) = 1. Find a formula that relates an + 2, an + 1, and