[ Pobierz całość w formacie PDF ]
n n
n=1 n=1
not absolutely convergent. We show using 6.29 that this series is still convergent, and so is
conditionally convergent.
Write an =1/n, so an >0, an+1
"
(-1)n+1
of Leibniz s theorem are satisfied, and so the series is convergent.
n
n=1
"
6.31. Proposition (Re-arranging an Absolutely convergent Series). Let an
n=1
be an absolutely convergent series and suppose that {bn} is a re-arrangement of {an}. Then
"
bn is convergent, and
n=1
" "
bn = an.
n=1 n=1
64 CHAPTER 6. INFINITE SERIES
Proof. See next year, or (Spivak 1967); the point here is that we need absolute convergence
before series behave in a reasonable way.
Warning: It is not useful to re-arrange conditionally convergent series (remember the
rearrangement I did in section 1.1). There is a result which is an extreme form of this:
"
Pick x " R, and let an be a conditionally convergent series. then there is
n=1
"
a re-arrangement {bn} of {an} such that bn = x!
n=1
In other words, we can re-arrange to get any answer we want!
6.5 An Estimation Problem
This section shows how we can use a lot of the earlier ideas to produce what is often wanted
in practice results which are an approximation, together with an indication of how good
the approximation is.
Find how accurate the approximation obtained by just using the first ten terms
"
1
is, to .
n2
n=1
Again we are going to use geometrical methods for this. Our geometric statement follows
from the diagram, and is the assertion that the area of the rectangles below the curve is less
than the area under the curve, which is less than the area of the rectangles which contain
the curve.
y =1/x2
N N+1 N+2 M-1 M
Figure 6.3: An upper and lower approximation to the area under the curve
Writing this out geometrically gives the statement:
M M-1
M
1 dx 1
d" d"
n2 N x2 n2
n=N+1 n=N
We can evaluate the middle integral:
M
M
dx 1 1 1
= - = - .
x2 x N M
N
N
6.5. AN ESTIMATION PROBLEM 65
For convenience, we define
" N
1 1
S = and SN = .
n2 n2
n=1 n=1
We can now express our inequality in these terms:
1 1
SM - SN d" - d" SM-1 - SN-1
N M
Next, let M ’! ", so SM ’!S, and 1/M ’! 0. We have
1
S - SN d" d" S - SN-1
N
Replacing N by N + 1, gives another inequality, which also holds, namely
1
S - SN+1 d" d"S-SN,
N +1
and combining these two, we have
1 1
S - SN+1 d" d"S-SN d" d"S-SN-1.
N +1 N
In particular, we have both upper and lower bounds for S - SN, as
1 1
d"S-SN d" .
N+1 N
To make the point that this is a useful statement, we now specialise to the case when
N = 10. Then
1 1 1 1 1 1
d" S - S10 d" or 0 d" S - S10 + d" - = .
11 10 11 10 11 110
Our conclusion is that although S10 is not a very good approximation, we can describe the
error well enough to get a much better approximation.
66 CHAPTER 6. INFINITE SERIES
Chapter 7
Power Series
7.1 Power Series and the Radius of Convergence
In Section 5.6, we met the idea of writing f(x) =Pn(x) +Rn(x), to express a function in
terms of its Taylor polynomial, together with a remainder. We even saw in 5.30 that, for
some functions, the remainder Rn(x) ’! 0 as n ’! "for each fixed x. We now recognise
this as showing that certain series converge.
We have more effective ways of showing that such a series converges we can use that
ratio test. But note that such a test will only show that a series converges, not that it
converges to the function used to generate it in the first place. We saw an example of such
a problem in the Warning before Example 5.36.
To summarise the results we had in Section 5.6,
7.1. Proposition. The following series converge for all values of x to the functions shown:
x2 x2 xn
ex =1 +x+ + + . . . +. . .
2! 2! n!
x3 x5 x2n+1
sin x = x - + + . . . +(-1)n+1 +. . .
3! 5! (2n +1)!
x2 x4 x2n
cos x =1 - + + . . . +(-1)n + . . .
2! 4! 2n!
x3 x5 x2n+1
sinh x = x + + + . . . + +. . .
3! 5! (2n +1)!
x2 x4 x2n
cosh x =1 + + + . . . + +. . .
2! 4! 2n!
These are all examples of the subject of this section; they are real power series, which
we can use to define functions. The corresponding functions are the best behaved of all the
classes of functions we meet in this course; indeed are as well behaved as could possibly
be expected. We shall see in this section that this class of functions are really just grown
up polynomials , and that almost any manipulation valid for polynomials remains valid for
this larger class of function.
[ Pobierz całość w formacie PDF ]