Math assignment

Week 7:

10.1(12, 18, 20)

In Exercises 7 – 12, write a general term for the sequence, if one exists, assuming a domain starting with n = 1.

12) 3, 8, 15, 24, 35, 48, …

In Exercises 13 – 18, compute the sums indicated.

18)

In Exercises 19 – 26, find the sum if the series converges.

20)

10.2(2, 6, 10, 14, 20)

For Exercises 1 – 8, find the Taylor polynomial of degree n approximating the given function near x = 0. Using a graphing utility, sketch the given function and the Taylor approximation on the same coordinate system.

2)

6)

For Exercises 9 – 16, find the Taylor polynomial of degree n near x = a for the given n and a.

14)

20) Construct the Taylor polynomial of degree 3 centered at x = 2, for the function . Use this polynomial to estimate a value for . Compare this to the value given by your calculator.

10.3(2, 4, 6, 8, 24)

2) Use the Taylor series for and substitution to obtain a power series for

4) Since arctan we have arctan

a) Substitute your answer to 3b. into the integral and integrate term-by-term to get a power series expansion for the arctangent function.

b) The interval of converges for the geometric series is (-1, 1). This means the interval of convergence for the solution to 4a. is also expected to be (-1, 1). However, in this case the interval includes the endpoint x = 1. Since the angle whose tangent is 1 is the angle arctan (1) = . Thus π= 4 arctan (1). Substitute x = 1 into your power series for 4(a) and get a numerical representation for πas an infinite series (due originally to Leibniz).

6) Replace the integral in Exercise 5 with and repeat the four steps.

8) In engineering, the hyperbolic sine function, abbreviated sinh, is defined by sinh(t) = .

a) Calculate the first four derivatives of sinh(t) and determine a degree 4 Taylor polynomial for y = sinh(t).

b) Combine the known Taylor series for and get a Taylor series for sinh(t). Compare your results with part (a).

In problems 23 – 26, find a Taylor series polynomial of degree at least four which is a solution of the boundary value problem.

24)