MathDB

Let

a

and

b

be real numbers with

a<b,

and let

f

and

g

be continuous functions from

[a,b]

(0,\infty)

such that

\int_a^b f(x)\,dx=\int_a^b g(x)\,dx

but

f\ne g.

For every positive integer

n,

define

I_n=\int_a^b\frac{(f(x))^{n+1}}{(g(x))^n}\,dx.

Show that

I_1,I_2,I_3,\dots

is an increasing sequence with

\displaystyle\lim_{n\to\infty}I_n=\infty.

A3