BC CALC 3
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Quiz #5
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#1(6 pts) Find the interval of convergence for the following power series. Show all steps
clearly. Check endpoints!


n 1
2n ( x  2) n
n  3n
#2(8 pts) Give the interval of convergence for each power series. No work necessary, though
partial credit may be given.

a.

n 0

c.

n 0
xn
n
(1)n ( x  4)n
n2  5n

b.

n 1

d.

n 1
( x  2) n
n  3n
( x  2) n
n!

#3(6 pts) Consider the power series
 a ( x  b)
n 0
n
n
.
a.
If this power series converges only for –17 ≤ x ≤ 13, determine its radius of
convergence and the value of b.
b.
Suppose it is known that b = 1 and that for this value of b, the power series
converges for x  4 and diverges for x  9 . What is the minimum and
maximum radius of convergence for this series?
#4 (6 pts) Suppose P2 ( x)  a  b( x  3)  c( x  3) 2 is the second degree Taylor polynomial
for f centered at x  3 . Determine the signs (positive, negative or zero) of a,b, and c if the
graph of f is shown below:
#5 (6 pts) The Maclaurin polynomial of degree 100 for some function f is given by
P100 ( x)  1  3  2 x  4  3 x 2  5  4 x3  6  5 x 4  7  6 x 5  8  7 x 6  9  8 x 7 
 102 101x100
a. Find f (0) .
b. Find f (50) (0) .
#6(8 pts) If f  x   1  x  3 , find the fourth degree MacLaurin series for f  x  .
5
#7(2 pts) Find all x which satisfy:


n 0
n 0
 2 x 2 n  3 x 3n
#8(2 pts) If p is a constant, find the interval of convergence for the power series

p  ( p  1)  ( p  2)
 ( p  (n  1))  x n
convergence set for the series 
.
n!
n 0