Math 1502

Quiz #2
Feb 1
Tom Morley

Open Book and Notes.  Carefully explain your proceedures and answers. Calculators allowed, but answers mush be exact.

Problem 1

a. Find a series for ln(1-x)  (look it up).

b. Find a series for ln(1-)

c. Find a series for  sin(2x)

d.  Find a series for sin(2 x)

e. find the limit as x-->  0    of

Ans

b: Just stubstitute. Everywhere you see an x put an :

c: Start with the series for sin(x), and everywhere you see an x, put an :

d: Multiply the above by :

Look at ratio of leading terms /= -1/2.

Problem 2 (10 points)

Eddy  Merckx Needs to compute an integral. He suggests the folling steps:

a. From the formula for Taylor series with error bound for   (Valid for |x| ≤ 1):

|    - ( 1 + x + +   + ... +   ) | ≤  ,

Derive a  formula for Taylor series with error bound for

b.  From the above derive a  formula for Taylor series with error bound for  

c. From the above derive a formula for Taylor series with error bound for

b: Now multiply the answer to a: by :



and if you like, simplify:

Integrate b: term by term: