Chapter 10.3, Problem 33E

The formula for the amount A in an investmentaccount with a nominal interest rate r at any timet is given by A(t)=a(e)rt, where a is the amount ofprincipal initially deposited into an account thatcompounds continuously. Prove that the percentageof interest earned to principal at any time t can becalculated with the formula I(t)=ert1.

Let f(x)=e(x-1)^2-1/(x-1)2  for x ≠ 1 and f(1)=1 a. Write the first four nonzero terms and the general term of the Taylor series for e(x-1)^2 about x=1 b. Use the series found in (a) to write the first four nonzero terms and the general term for the Taylor series for f about x=1 c. Determine the interval of convergence for the series given in (b). d. Use the series for f about x=1 to determine if the graph of f has any points of inflection.

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?