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5.1.1. The series may either converge or diverge when |x x0| = p.
0 . Series _ .
Series converges ->+-Series _
diver9es absolutely diverges
- p x0 + p
Series may Ó converge or diverge
FIGURE 5.1.1 The interval of convergence of a power series.
Determine the radius of convergence of the power series
(x + 1)n
We apply the ratio test:
(x + 1)n+1
(n + 1)2^ (x + 1)
|x + 1|
n^TO n + 1
|x + 1|
Thus the series converges absolutely for |x + 11 < 2, or 3 < x < 1, and diverges for | x + 1 | > 2. The radius of convergence of the power series is p = 2. Finally, we check the endpoints of the interval of convergence. At x = 1 the series becomes the harmonic series
which diverges. At x = 3 we have
g (3 + 1)n
which converges, but does not converge absolutely. The series is said to converge conditionally at x = 3. To summarize, the given power series converges for 3 < x < 1, and diverges otherwise. It converges absolutely for 3 < x < 1, and has a radius of convergence 2.
If an (x x0)n and bn(x x0)n converge to f (x) and g(x), respectively,
for |x x0| < p, p > 0, then the following are true for |x x0| < p.
6. The series can be added or subtracted termwise and
f (x) ą g(x) = (an ą bn)(x x0)n.
Chapter 5. Series Solutions of Second Order Linear Equations
7. The series can be formally multiplied and
f (x )g(x) =
J2an(x x0 )n
J2bn (x - x0)n
= J2 cn (x - x0)n
where cn = a0bn + a^n_ 1 + + anb0. Further, if g(x0) = 0, the series can be formally divided and
f (x) g(x)
= YI dn (x - x0)n
In most cases the coefficients dn can be most easily obtained by equating coefficients in the equivalent relation
Yan (x x0)n =
Ydn (x - x0)n
to / n
Ybn (x - x0)n
Hdkbn-k) (x - x0)n
Also, in the case of division, the radius of convergence of the resulting power series may be less than p.
8. The function f is continuous and has derivatives of all orders for |x x01 < p. Further, f, f", can be computed by differentiating the series termwise; that is,
f(x) = a1 + 2a2(x x0) +-------+ nan (x x0)n1 +--------------
= ? nan (x x0)n1, n = 1
f'(x) = 2a2 + 6a3(x x0) +-+ n(n 1)an (x x0)n2 +---------------------
= Yn(n 1)an(x x0)n 2,
and so forth, and each of the series converges absolutely for |x x0| < p.
9. The value of an is given by
an = ~~n\
The series is called the Taylor1 series for the function f about x = x0.
10. If J] an (x x0)n = bn (x x0)n for each x, then an = bn for n = 0, 1,
2, 3, In particular, if YI an(x x0)n = 0 for each x, then a0 = a1 = ŚŚŚ =
an = ŚŚŚ = 0.
1Brook Taylor (1685 -1731) was the leading English mathematician in the generation following Newton. In 1715 he published a general statement of the expansion theorem that is named for him, a result that is fundamental in all branches of analysis. He was also one of the founders of the calculus of finite differences, and was the first to recognize the existence of singular solutions of differential equations.
5.1 Review of Power Series