(^342) Functions, graphs, and numbers
N
where C = Z Isk - s4. If we choose Ni such that Ni > N and C/n < e/2
k=1
when n > N, then we will have
(6) I Mn - sl < e
when n > Ni. This proves (3). Remark: It often happens that the limit in
(3) exists when the limit in (2) does not exist. In case u1 + u2 + u3 + is
a series having partial sums s1, s2, and arithmetic means M1, M2, M3,
such that (3) holds, we can write
(7) S = Cil{u1 + u2 + u3 +. ..}
and say that the series is evaluable to s by the method of arithmetic means or by the
Cesaro method of order 1.
8 Supposing that n is a positive integer, sketch a graph of the function f
for which fn(x) = n2x when lxI < 1/n and fn(x) = l/x when IxI > I/n. Show
that f,, is continuous over El. Show that
limfn(x) = g(x),
where g(O) = 0 and g(x) = 1/x when x s 0. Hint: Consider separately the
cases in which x = 0, x > 0, and x < 0.
9 Using the notation of the preceding problem, let
ui(X) = fl(x)
U2(X) = f2(X) - fl(x)
U3(X) = f3(x) - f2(X)
u4(x) = f4(x) - f3(x),
etcetera, so that uk(x) = fk(x) - fit_1(x) when k = 2, 3, 4,. Show that
each function un is continuous over El and that
I uk(x) = g(x)
k=1
Remark: It is sometimes necessary to be sophisticated enough to know that a
series of continuous functions may converge to a discontinuous function. More-
over, we should be tall enough to peer over the wall of our garden and observe
that a series u1(x) + u2(x) + of functions having partial sums f1(x),
f2(x), is said to converge uniformly over a set E to f(x) if to each positive
number a there corresponds an integer N such that lfn(x) - f(x)l < e whenever
n >t N and x is in E. The following theorem is proved in advanced calculus.
If a series u1(x) + u2(x) +. of continuous functions converges uniformly over
E to f(x), then f must be continuous over E.
(^10) Starting with positive numbers a1 and b1 for which a1 < b1, let sequences
a1i a3, a3, and b1, b2, b3, be defined recursively by the formulas
()^1 an}1 '° , bn+1 =an + bn^2
lu
(lu)
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