Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

98 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES
(f) Ratio Test:
(i) If lim I an+l ) = L < 1, then the series f an is absolutely convergent (and therefore convergent).
n-oo an no=l
(. ") If 1" I an+1 I L 1 1" I O.n+l I h h . ~ . d"
ll un - - = > or 1m -- = oo, t en t e senes L- a n IS Ivergent.
n-oo an n--+oo an n=l
(iii) If lim I an+l I = 1, the Ratio Test is inconclusive; that is, no c~nclusion can be drawn about the convergence or
n-+co an
(g) Root Test:
divergence of 2:: an.
(i) If lim \li(i:j = L < 1, then the series 2::;:'= 1
a,. is absolutely convergent (and therefore ~nvergent).
n-co
(ii) If lim \li(i:j = L > 1 ~r
n-.oo
(iii) If lim
11.--+00
lim ~ = oo, then the series 2::;:'= 1
an is divergent.
n--+oo
~ = 1, the Root Test is inconclusive.
6. (a) A series L an is calledab~olutely convergent if the series of absolute values 2:: I ani is convergent.
(b) If a series L an is absolutely convergent, then it is convergent.
(c) A series L:al' is called conditionally convergent if it is convergent but not absolutely convergent.
7. (a) Use (3) in Section 11.3.
(b) See Example 5 in Section 11.4.
(c) By adding terms until you reach the desired accuracy given by the Alternating Series Estimation Theorem.
00
8. (a) 2:: en(x - a)"
n=O
00
(b) Given the power series L en(x- a)", the radius of convergence is:
n=O
(i) 0 if the series converges only when x = a
(ii) oo if the series converges (or all x, or
(iii) a positive number R such that the series converges if lx - al < Rand diverges if lx- al > R.
(c) The interval of convergence of a power series is the interval that consists of all values of x for which the series converges.
Corresponding to the cases in part (b), the interval of convergence is: (i) the single point {a}, (ii) all real numbers, that is,
the reaL number line ( -oo, oo ), or (iii) an interval with endpoints a - R and a + R which can contain neither, either, or
both of the endpoints. In this case, we must test the series for ·convergence at each endpoint to determine the interval of
convergence.
9. (a), (b) See Theorem 11 .9.2.
n
f(i)(a)
10. (a) Tn(x) = L - .- 1
- (x - a)'
i =O ~.
oo j (n){ )
(b) I: __ a_ (x - a)"
n=O n l
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