Open Book and Notes. Carefully explain your proceedures and answers. Calculators allowed, but answers mush be exact.
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
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:
Converted by Mathematica
February 1, 2000