264 CHAPTER 5 Series and Differential Equations
(a) IfG(x) =∫x
0 f(t)dt, thenF′(0) =G′(0).
(b) Show thatGis anoddfunction, i.e.,G(−x) =−G(x).
(c) Use integration by parts to show that if 0< y < x, then
∫x
y f(t)dt =∫x
yt^2 sin(1/t)dt
t^2
= x^2 cos(1/x)−y^2 cos(1/y) +∫x
y^2 tdt≤^3 x(^2).
(d) Using part (c), show that for allx, |G(x)|≤ 3 x^2.
(e) Conclude from part (d) thatG′(0) = 0.)
5.2 Numerical Series
Way back inAlgebra IIyou learned that certaininfinite seriesnot
only made sense, you could actually compute them. The primary (if not
the only!) examples you learned were theinfinite geometric series;
one such example might have been
3 +
3
4
+
3
16
+
3
64
+···=
∑∞
n=03
4 n.
Furthemore, you even learned how to compute such infinite geometric
series; in the above example since the first term is a = 3 and since
theratioisr=^14 , you quickly compute the sum:
3 +
3
4
+
3
16
+
3
64
+···=
∑∞
n=03
4 n=
a
1 −r=
3
1 −^14
= 4.
Perhaps unfortunately, most infinite series are not geometric but
rather come in a variety of forms. I’ll give two below; they seem similar
but really exhibit very different behaviors:
Series 1: 1 +