(^604) Series
3 Prove that if s is a constant, then the series
lax + 28x2 + 38x3 + 48x4 + ...
converges when lxl < 1 and diverges when IxI > 1.
4 When x = 1, the ratio test does not tell whether the series of the preceding
problem is convergent. Using this hint, give an example of a divergent series
fun for which
(1) limun+i+I= 1,
n-4 m un
and then give an example of a convergent series for which (1) holds.
5 The nth term u" of the series
(1 i)2 + (21)2 + (3')2 +
is (2n)!/(n!)2. Prove that
lim u+1 = lim(2n + 2)(2n + 1)--4
n_ m nn n-..o (n + 1)(n + 1)and hence that the series is divergent.
6 What information does the ratio test give about convergence of the series(1!)2 (21)2 (3!)2 (4!)2
2! 4Fx2}
6 x3 + 81 x4 +...?
dns.: The series converges when IxI < 4 and diverges when (xI > 4. The ratio
test gives no information when x is 4 or -4.(^7) Supposing that m 5A a, show that the series in
(^11) x-a (x-a)2 (x-a)3
x - m m- a (m-a)2 (m -a)3 (m - a)4
is a geometric series and that it converges to 1/(x - m) when Ix - al < Im - al.
(^8) Supposing again that m 34 a, show how the calculation
(^1) -1 -1 _ -1 1
X I n 1 - x-a
m - a
can be used to obtain the formula of the preceding problem when Ix - al <
Im - al. Hint: We must always know that the geometric series
1+r +r2+ra+ ...
converges to 1/(1 - r) when Irl < 1, and we must sometimes be wise enough to
start with 1/(1 - r) and write the geometric series that converges to it when
Irl<1.