Quiz 7 - Math XXXXXXXXXX)
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• This quiz is due at 11:59 PM (Eastern Time) on Saturday, December 10.
***********************
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1
Quiz 7 - Convergence Tests of Series
1. Using the alternating test, determine whethe
∞∑
n=0
sin
(
2n+1
2
π
)
1 +
√
n
is convergent or diver-
gent.
2. Chapter 10-7, Problem 36.
3. Chapter 10-8, Problem 20.
4. Chapter 10-9, Problem 38.
5. Chapter 10-10, Problem 32.
2
xX x x ® k. xT x x x
sin) =x XXXXXXXXXXI7 +.= 3EDEFETT. = Lex IT XXXXXXXXXX +
=)
Find a power series for sin( x?) 34. Find a power series for sin( 2x )
. Find a power series for ¢™ 36. Find a power series for ¢™*)
Find a power series for cos( x ) 38. Find a power series for cos( x?)
1
Represent the integral as a merical series: J sin) ax
0
Represent the integral as a numeric series; Jo
0
n(x) a. fim SH-X
x 0 x
109 Representing Functions as Power Series. Contemporary Calculus
Practice Answers
Practie 1: meric Sess Font 540 nit
ey
1 Pa
(2) Replacing "x" with "x> we have he 3 for IX 1<1 or Ixl<1
16. Foxe”
21 5 nlx" 5 nlx +2)"
n=l n=1 n=0
In problems 25 — 34, the letters "a" and "b" represent positive constants. Find all values of x for which each
given power series converges.
= x-a)
27. n
n=1
31 3 (ax=b)" 32 3 (ax +b)”
n=1 n=1
108 Power Series Contemporary Calculus
22. J cos(x") dx and J cos(x”) dx sin(x-)dx and J sin(x) dx
ud Dam I Ha J ama Sa
26 J moa aa J moa 2d incom ana J Fano
In problems 28 — 35, use the series representation of these functions 10 calculate the limits.
0. jm 100
xs0 x1
3 0
x0 x
x-sin(x)
lim
0 x
1
Use the series for ¢* and ¢™* 10 write a series representation for cosh(x) = 7 (&" +¢
(The function "cosh" is called the hype
olic cosine function.)
Use the series for ¢* and ¢™* to write a series representation for sinh(x)
(The function "sinh" is called the hype
olic sine function.)
10.10 Taylor and Maclaurin Series “ontemporary C
38. Show that D( series for cosh( x ) in problem 36 ) is the series for sinh( x ) in problem 37.
“The Ratio Test is commonly used with series that contain factorials, and factorials are also going 10 become.
more common in the next few sections. Problems 31-40 ask you to simplify factorial expressions in
order to get ready for that usage.
(Definitions: 0! = 1,11 = 1,2! = 11222, 31 = 12:3 = 6,4! = 24, and, in general, n! = 12:3 .. {(n-1)'n )
a 1 : 1
31. Show that TT = mT . Show that G3 = ITE)
n! 1 . (1
33. Show that Tu3)T = ETE DE 34. Show that
G a1! 1
35. Show that {ni T)T = (sD XXXXXXXXXX)"..2n)
(20)!
Qn 1 1
37. Show that oh = mT - Show that T3(py Dy) = Tanned)
107 Absolute Convergence and the Ratio Test “ Calculus
”
39. Show that jr = a 40. Forn>0, which is larger, 7! or 241% 5
o nil
In problems XXXXXXXXXX, (a) determine the value of L= lim | <= | for the given series, and
nx
Fr