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Unformatted text preview: ber x in the interval a, b
and choose an integer N such that
Ý kJ , y n
nN 4 . Using the fact that f is continuous at each of the numbers y n for n N, choose 0 such that
whenever n N and t y n  we have
.
f t f yn 
2 var , a, b
Now suppose that
P x 0 , x 1 , , x p
x j 1 , x j for each j, and define g to be the
is any partition of a, b for which P and that t j
step function that takes the value f x j at each number x j and that takes the constant value f t j in
each interval x j 1 , x j . We see that Þa b gd Þa b Ý Ý g y n J , y n fd
n1 f y n J , y n
n1 N Ý N g y n J , y n
n1 f y n J , y n
n1 Ý g y n J , y n
nN1 f y n J , y n
nN1 Ý N g y n f y n J , y n 2 n1 kJ , y n
nN1 N
n1 2 var , a, b J , y n 2 3. By combining the preceding two exercises, obtain an analog of Exercise 2 that does not require the
assumption that varies discretely on a, b .
318 The desired extension to the case in which is an arbitrary increasing function follows at once
when we split into its continuous and discrete parts. 12 Infinite Series
Some Elementary Exercises on Series
1. Find the nth partial sum of the series
n3
.
n n1 n3
Deduce that this series is convergent and find its sum. Solution: Given any positive integer j we have
j3
j j1 j3 2
j1 1
j 1
j3 and therefore
n
j1 j3
j j1 j3 n 1
j 2
j1
j1
n 1 1
j3
n 3 n 2
j
j2 1
j j1 j4 22 2
2
3
n1
22
1
1
2
3
1
2
as n
2. 1
j 1
1 1
2 1
3 1
3 1
n1 1
n2 1
6 Ý.
xn. a. Find the derivative of the nth partial sum of the series
If n is any positive integer and x 1 then
n xj
j1 xn
.
x x1
1 Differentiating we obtain
n jx j 1 1 xn j1 xn nx n nx n1 1
1 x2 nx n 1
1 x2 nx n 1 . Deduce that if x  1 we have b. Find the nth partial sum of the series Ý nx n 1 nx n 1 n1 1
1 x 2 . In order to deduce the identity
Ý n1 we need to take the limit as n 1
1 x 2 Ý of each side of of the identity
n jx j 1 1 j1 319 xn
1 nx n nx n1
x2 x 1
n3 and for this purpose we need to know that
lim nx n 0 nÝ whenever x  1. Later in this chapter, we shall see some simple ways of finding the latter limit.
Perhaps the simplest way to find it right now is to use L’Hôpital’s rule. Suppose that x  1. To
show that nx  n 0 as n Ý we use the fact that
n
nx  n
1/x  n
We define c log 1/x  . Note that c 0 and
nx  n n
e cn
n
and the fact that nx 
0 as n Ý follows from the fact that
lim tct 0
tÝe
which follows easily from L’Hôpital’s rule.
An alternative to L’Hôpital’s rule is to give the students the assignment of proving the inequality
22
e ct 1 ct c t
2
for all t 0. This inequality follows easily from the mean value theorem and from it we obtain
the above limit easily.
3. Given that x  1 and that
n sn 1 x 2j 3j
j1 for every positive integer n, obtain the identity
3x 4
s n 1 x 2 2x 2 3x 2n4 3n 1 x 2n2
1 x2 x 2 n 4 and deduce that
Ý 3x 4
1 x2 1 x 2j 2x 2 3j
j1 We observe that
n sn...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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