Use a geometric series to represent each of the given functions as a power series about x=0, and find their intervals of convergence. a. f(x)=5/(3-x) b. g(x)= 3/(x-2)

Find the coefficient of x° in the power series of this function f(x) = (1-2.x)2

Find the power series for each function and identify its interval of convergence. f(x) = 1 – x2 1.

Let f(x) = 1 + x 1 X Find the power series representation for the function f(x) by completing the following steps: a. First, express the fraction 11 as a power series. = X b. Now, express the fraction as a power series. 1-x 1+x x c. The function f(x) 1-x 1-x + 1 is the sum of the two series from parts (a) and (b). Express the function f(x) as a power series. d. What is the interval of convergence and the radius of convergence for this power series?

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Observe the function X f(x) = (1+2x)2 In order to find the power series for this function, complete the following steps: 1 1-x a. Start with the series Σ. Replace x with (−2x) in this series and k=0 write the corresponding power series for = 1 1+2x b. Take derivative of the series from part (a) above and relate it to the power series for the function 1 (1+2x)2· c. Multiply both sides of the resulting series from above with x, and obtain the series for Write the first four non-zero terms of this series. X (1+2x)2 d. What is the radius of convergence for this series? What is the interval of convergence?

1. Find the power series for f(x) = 1-BX

Find the first four nonzero terms of the binomial series expansion of the function f (x) = (1 +x10)°. Simplify your answer. +.. + +

An individual is given a dose of Q units of a certain drug. the amount that remains in the bloodstream at the end of t minutes and Qe−ct, with c > 0. Suppose that the same dose is administered at successive intervals of T minutes. (a) Using series, express the amount A(k) of the drug in the bloodstream immediately after the K-th dose.(b) Find an upper bound for the amount of drug in the bloodstream after an arbitrary number of doses.(c) Find the shortest time between doses that ensures that A(k) does not exceed a certain level M, M > Q.

For the function ln(5x+11), (a) write a power series representation of the function, (b) find the center and radius of convergence of the power series from (a), and (c) find the interval of convergence of the same power series.

Find the power series for f(x) = (sin x)/x. Show your work. Your final answer should be in summation notation. You are required to show your work and/or provide an explanation for credit.

2. (15 pts) Write the Maclaurin series for the function f(x) = sinx. Use it to write the Maclaurin series for g(x) = sin. Express find as a series. Write it in sigma notation as well.

Find a power series representation for the function below, and determine the interval ofconvergence. f(x) = (x)/(π−x^2)