What-are-the-neccesary-and-sufficient-conditions-so-that-n-0-f-n-x-dx-n-0-f-n-x-dx-

Question Number 4133 by prakash jain last updated on 29/Dec/15

What are the neccesary and sufficient

conditions so that

\int_{- \infty}^{+ \infty} [\underset{n = 0}{\sum^{\infty}} f (n, x)] d x = \underset{n = 0}{\sum^{\infty}} [\int_{- \infty}^{+ \infty} f (n, x) d x]

Commented by prakash jain last updated on 29/Dec/15

Thanks .

Commented by Yozzii last updated on 29/Dec/15

I f t h e t e r m s o f s = \underset{n = 0}{\sum^{\infty}} f (n, x) a r e

c o n t i n u o u s o n [a, b] a n d i f t h e s e r i e s

i s u n i f o r m l y c o n v e r g e n t o n [a, b], t h e n

a) T h e s u m i s c o n t i n u o u s;

b) T h e s u m c a n b e i n t e g r a t e d t e r m b y

t e r m \Rightarrow \int_{a}^{b} [\underset{n = 0}{\sum^{\infty}} f (n, x)] d x = \underset{n = 0}{\sum^{\infty}} [\int_{a}^{b} f (n, x) d x] .

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