Math 1320-6 Lab 5
Name:
T.A.: Kyle Steffen
25 February 2016
uNID:
Instructions and due date:
• Due: 3 March 2016 at the start of class.
• For full credit: Show all of your work, and simplify your final answers.
• Work together! However, your work should be your own (not copied
from a group member).
1. Give an example of a power series with the following radii of convergence: (a) R = 0; (b) R = 3;
(c) R = ∞. Justify your answer.
(a) R = 0
(b) R = 3
(c) R = ∞
Math 1320-6
Lab 5 - Page 2 of 4
2. Consider the function f (t) = (arctan t)/t.
Rx
(a) Compute
a power series expansion of I(x) = 0 f (t)dt. Write your answer in the form
P∞
2k−1 for an appropriate choice of a . Compute the radius of convergence of the
k
k=1 ak x
resulting power series expansion.
P
(b) Let In (x) = nk=1 ak x2k−1 be the n-th partial sum of the series in part (a) (i.e. it consists
of the first n terms of the power series). Define En to be the error of approximating I(1)
by In (1): En = |I(1) − In (1)|.
What is the minimum number of terms N required so that EN < 0.01? What if we require
EN < 0.001? EN < 0.000001 = 10−6 ? (Hint: Use the Alternating Series Estimation
Theorem. Also: All three of your answers should be positive integers.)
Math 1320-6
Lab 5 - Page 3 of 4
3. Consider the following graph (in blue) of a function f (x). Answer the following questions, and
justify your answer with a short sentence or two.
3
2
y
1
0
−1
−2
−3
−5 −4 −3 −2 −1
(a) Is
0
x
1
2
3
4
5
3
1
1
5
− x + x2 + x3 − x4 + · · · the Maclaurin series of f (x)?
4
2
2
8
5 5
5
5
1
(b) Is − + (x + 1) + (x + 1)2 + (x + 1)3 − (x + 1)4 + · · · the Taylor series of f (x) at
7 9
4
12
7
the point x = −1?
8 8
3
15
3
(c) Is + (x − 3) + (x − 3)2 + (x − 3)3 − (x − 3)4 + · · · the Taylor series of f (x) at
5 9
18
31
11
the point x = 3?
Math 1320-6
Lab 5 - Page 4 of 4
4. The following limits represent some derivative of f (x). Use Taylor series to determine which
derivative it is.
1
(a) lim 2 [f (x + h) + f (x − h) − 2f (x)]
h→0 h
1
[f (x + h) − f (x − h)]
h→0 2h
(b) lim