Hi, I wonder if this hypothesis is true:(adsbygoogle = window.adsbygoogle || []).push({});

Let [tex]f_n[/tex] be an arbitrarily chosen n'th anti-derivative of the function [tex]f_0[/tex]. Similarly, let [tex]g_n[/tex] be the n'th derivative of the function [tex]g_0[/tex].

Now, [tex]\int^b_a f_0 g_0 \rm{d}x=[f_1g_0]^b_a-\int^b_a f_1g_1 \rm{d}x=[f_1g_0-f_2g_1+...]^b_a+(-1)^n \int^b_a f_{n+1}g_n \rm{d}x[/tex].

hypothesis:

If [tex]\lim_{n \to \infty} f_{n+1}g_n =0[/tex] for all continous intervals of and never diverges. Then

[tex]\int^b_a f_0 g_0 \rm{d}x = [\sum^{\infty}_{n=0} (-1)^n f_{n+1}g_n]^b_a[/tex]

This seems intuitively correct, but I wonder how to prove it.

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# Infinite series by integration by parts

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