Math 143: Calculus III Practice Midterm II
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Math 143: Calculus III
Practice Midterm II
November 8th, 2022
1. (15 points) For each of the following series, either use the Alternating Series Test (with justification) to show that it converges or explain why the Alternating Series Test does not apply.
- cos(nπ)en
n!
- (-1)n cos(nπ)nn
n=1
- (-1)n
n2 + 5
n=1
2. (12 points) Determine whether the following series converges absolutely, converges only conditionally, or diverges. Name any test you use and justify its use.
- (-1)n
3. (13 points) Determine whether the following series converges absolutely, converges only conditionally, or diverges. Name any test you use and justify its use.
- sin(n)
2
4. (15 points) Find the radius and interval of convergence of the following power series.
- (-5)n (x - 3)n
(n - 2)3/24n .
5. (15 points)
(a) Consider the function f (x) = ln(2x). Find a power series expansion of f (x) about x = 3.
(b) Use the ratio test to find the radius and interval of convergence of the series you found in (c). No credit will be given for solutions not using the ratio test.
6. (15 points)
(a) Find the Maclaurin series expansion of the function
z
f (x) =
write out the first four nonzero terms, and express the series in sigma notation.
(b) What is the value of f(10) (0)?
(c) What is the value of f(11) (0)?
(d) What is the value of lim f (x)?
x→0
7. (15 points) Write out the first three terms and then find the sum of each of the following series. Your table of Maclaurin series expansions might be helpful.
-
(a) 10n =
(b) =
(c) =
n=1
Common Taylor series centered at x = 0:
Function |
Taylor Series |
Initial Terms |
Converges for |
1 1 - x |
- xn n=0 |
1 + x + x2 + x3 + x4 + . . . |
-1 < x < 1 |
1 1 + x |
- (-1)nxn n=0 |
1 - x + x2 - x3 + x4 - . . . |
-1 < x < 1 |
北 |
- xn n! n=0 |
1 + x + + + + . . . |
All x |
sin(x) |
- (-1)n (2n + 1)! n=0 |
x - + - + . . . |
All x |
cos(x) |
- (-1)n (2n)! n=0 |
1 - + - + . . . |
All x |
tan_1 (x) |
- (-1)n 2n + 1 n=0 |
x - + - + . . . |
-1 < x < 1 |
ln(1 + x) |
- (-1)n_1 n |
x - + - + . . . |
-1 < x < 1 |
2022-11-04