MATH 112Z QUIZ 7 Solution
(a) Find the third-order Maclaurin polynomial T3 (x) for f (x) = (1 + x)1/2 and use it to
approximate (0.5)1/2 .
Solution (a). Compute the derivatives:
1
3
5
1
1
3
f (x) = (1 + x)1/2 , f (x) = (1 + x) 2 , f (x) = (1 + x) 2 , f (x
MATH 112Z QUIZ 6 Solution
1. Find the sum of the series
nxn ;
(a)
n=1
Solution. Dierenting the G-series, we obtain that
nxn1 = (
n=1
xn ) = (
n=0
1
1
)=
.
1x
(1 x)2
Thus, multiplying the above by x gives that
nxn =
n=1
x
.
(1 x)2
(b)
(1)n 2n
.
2n
n=0 6 (2
MATH 112Z QUIZ 5 Solution
1. Find the convergence set for the power series:
2n (x 3)n
;
n+1
n=0
(a)
(n!)xn .
(b)
n=1
Solution (a). Consider the ratio:
|an+1 |
2n+1 (x 3)n+1
n+1
n+1
=|
| n
2|x 3|
| = 2|x 3|
|an |
n+2
2 (x 3)n
n+2
as n . Thus by the Ratio
MATH 112Z QUIZ Solutions
Quiz 4
1. Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
(a)
(1)n
.
n=1 5n + 1
Solution. Let an = (1)n /(5n + 1) denote the n-th term of the series. Then
|an |
n
1
=
as n .
1/n
5n +
MATH 112Z QUIZ Solutions
Quiz 2
1. Explain why the Integral Test cannot be used to determine the convergence or divergence of
the series
cos2 n
.
2
n=1 1 + n
Solution: The corresponding function f (x) =
cos2 x
1+x2
is not decreasing on [1, ).
2. Use the I
MATH 112z MIDTERM 1 Solutions
Print Your Name (Last, First): Answer Key No books, notes, calculators, etc. Show all your work; Clearly indicate your answers for each part.
10-2-2013
1. (8 points) Determine whether the given series is convergent or diverge