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
![[Graphics:Images/index_gr_1.gif]](Images/index_gr_1.gif)
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
![[Graphics:Images/index_gr_5.gif]](Images/index_gr_5.gif)
![[Graphics:Images/index_gr_6.gif]](Images/index_gr_6.gif)
b: Just stubstitute. Everywhere you see an x put an
:
![[Graphics:Images/index_gr_8.gif]](Images/index_gr_8.gif)
![[Graphics:Images/index_gr_9.gif]](Images/index_gr_9.gif)
c: Start with the series for sin(x), and everywhere you see an x, put an
:
![[Graphics:Images/index_gr_11.gif]](Images/index_gr_11.gif)
![[Graphics:Images/index_gr_12.gif]](Images/index_gr_12.gif)
d: Multiply the above by
:
![[Graphics:Images/index_gr_14.gif]](Images/index_gr_14.gif)
![[Graphics:Images/index_gr_15.gif]](Images/index_gr_15.gif)
![[Graphics:Images/index_gr_16.gif]](Images/index_gr_16.gif)
Look at ratio of leading terms
/
= -1/2.
![[Graphics:Images/index_gr_19.gif]](Images/index_gr_19.gif)
![[Graphics:Images/index_gr_20.gif]](Images/index_gr_20.gif)
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
![[Graphics:Images/index_gr_31.gif]](Images/index_gr_31.gif)
b: Now multiply the answer to a: by
:
![[Graphics:Images/index_gr_33.gif]](Images/index_gr_33.gif)
and if you like, simplify:
Integrate b: term by term:
![[Graphics:Images/index_gr_36.gif]](Images/index_gr_36.gif)
Converted by Mathematica
February 1, 2000