Advanced High-School Mathematics

254 CHAPTER 5 Series and Differential Equations

=

∑n

i=0

(−1)if(2i)(x), n≥ 1 ,

and show that

(e) Fn(0) andFn(π) are both integers;

(f) Fn′′(x) +Fn(x) =f(x) (note that if i > 2 n, thenfn(i)(x) = 0);

(g)

d

dx

[Fn′(x) sinx−Fn(x) cosx] =f(x) sinx;

(h)

∫π

0 fn(x) sinxdxis an integer for alln≥1.

Finally, show that

(i) fn(x) sinx >0 when 0< x < π;

(j) fn(x) sinx <

πnan

n!

when 0< x < π;

Conclude from (j) that

(k)

∫π

0 fn(x) sinxdx <

πn+1an

n!

for alln≥1.

Why are (h) and (k) incompatible? (Note that limn→∞

πn+1an

n!

= 0.)

5.1.2 Improper integrals

There are two type ofimproper intergralsof concern to us.

(I.) Those having at least one infinite limit of integration, such as

∫∞

a

f(x)dx or

∫∞

−∞

f(x)dx.

(II.) Those for which the integrand becomes unbounded within the

interval over which the integral is computed. Examples of these

include

∫ 1

0

dx

xp

, (p >0),

∫ 3

1

dx

x− 2

.