AMT Homework 2 Supplement

Dr. Osborne 2010 The purpose of this supplement is to illustrate some of the ways in which expansions can be finessed and used to approximate certain integral expressions.

1. Find the first five nonzero terms of the Maclaurin expansion of 21( )

1 2f x

x x . What

is the radius of convergence of the full expansion? Hint: you do not need to find the general term in order to determine the radius.

2. Determine the first four nonzero terms of the Maclaurin expansion for 1( )

(1 )(2 )(3 2 )f x

x x x . What is the radius of convergence of the full expansion?

3. Determine the first four nonzero terms of the Maclaurin expansion for 22

( )1x

xf xe

. These take you to the eighth power of x, and you may find it easier to consider the coefficient of each term separately in order to organize your calculation. What is the radius of convergence of the full expansion?

4. Determine the first four nonzero terms of the Maclaurin expansion for ( ) cotf x x x ,

and use your results to determine the exact values of the Riemann Zeta function ( )s at s = 2, 4, and 6. What is the radius of convergence of the full expansion?

5. Find a Maclaurin expansion for the function 2( ) ln 2 3f x x to all orders in x. What

is the radius of convergence of this expansion?

6. Find a Maclaurin expansion for the function 30( ) ; 11x dtf x x

t to all orders in x.

Why is the restriction on x necessary for the expansion? Do you expect the integral to converge for x = 3? What about x = -3? Your expansion should converge conditionally at x = 1. Use the first 5 nonzero terms in the expansion of f (1) to determine an approximation

for f (1) including error bounds. The exact value of f (1) is ln 2

33 3

. Does this value lie within your bounds? How many terms in your expansion would you have to keep in order to guarantee an error less than 75 10 (six digits)? Would you say that the expansion converges quickly?

7. Find a Maclaurin expansion for the function

0

1 cos 2( )

x tf x dt

t

. What is its radius of convergence? Use your expansion to determine the value of f (1) correct to 6 decimal places. Explain how you know that the error is less than 75 10 .