Question Number 4133 by prakash jain last updated on 29/Dec/15
![What are the neccesary and sufficient conditions so that ∫_(−∞) ^( +∞) [Σ_(n=0) ^∞ f(n,x)]dx=Σ_(n=0) ^∞ [∫_(−∞) ^(+∞) f(n,x)dx]](https://www.tinkutara.com/question/Q4133.png)
$$\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{neccesary}\:\mathrm{and}\:\mathrm{sufficient} \\ $$$$\mathrm{conditions}\:\mathrm{so}\:\mathrm{that} \\ $$$$\int_{−\infty} ^{\:+\infty} \left[\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{f}\left({n},{x}\right)\right]{dx}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left[\int_{−\infty} ^{+\infty} {f}\left({n},{x}\right){dx}\right] \\ $$
Commented by prakash jain last updated on 29/Dec/15
$$\mathrm{Thanks}. \\ $$
Commented by Yozzii last updated on 29/Dec/15
![If the terms of s=Σ_(n=0) ^∞ f(n,x) are continuous on [a,b] and if the series is uniformly convergent on [a,b], then a)The sum is continuous; b)The sum can be integrated term by term ⇒∫_a ^b [Σ_(n=0) ^∞ f(n,x)]dx=Σ_(n=0) ^∞ [∫_a ^b f(n,x)dx].](https://www.tinkutara.com/question/Q4151.png)
$${If}\:{the}\:{terms}\:{of}\:{s}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{f}\left({n},{x}\right)\:{are}\: \\ $$$${continuous}\:{on}\:\left[{a},{b}\right]\:{and}\:{if}\:{the}\:{series} \\ $$$${is}\:{uniformly}\:{convergent}\:{on}\:\left[{a},{b}\right],\:{then} \\ $$$$\left.\mathrm{a}\right){The}\:{sum}\:{is}\:{continuous}; \\ $$$$\left.\mathrm{b}\right){The}\:{sum}\:{can}\:{be}\:{integrated}\:{term}\:{by} \\ $$$${term}\:\Rightarrow\int_{{a}} ^{{b}} \left[\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{f}\left({n},{x}\right)\right]{dx}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left[\int_{{a}} ^{{b}} {f}\left({n},{x}\right){dx}\right]. \\ $$