# Sequence of number

## Main Question or Discussion Point

given the sequence (power series) $$g(x)= \sum_{n\ge 0}a(n)(-1)^{n}x^{n}$$

if i define $$a(n)=\int_{n=0}^{\infty}dxf(x)x^{n}$$ (1)

if $$f(x)>0$$ on the whole interval $$(0,\infty)$$ , is the solution to (1) unique ?? , this means that the moment problem for a(n) would have only a solution.

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HallsofIvy
Homework Helper
How can we tell you if "the solution to (1)" is unique when there is nothing labeled (1)?

And what does an integral of f(x) have to do with a sum of g(x)?

Frankly, nothing here makes any sense.

i meant

is the solution to

$$a(n)=\int_{n=0}^{\infty}dxf(x)x^{n}$$

where a(n) is given but f(x) is unknown UNIQUE is f(x) is positive on the whole interval (0,oo) ??? , i mean if the integral equation

$$a(n)=\int_{n=0}^{\infty}dxf(x)x^{n}$$

has ONLY a solution provided f(x) is always positive, thanks.

I don't think your integral should start at n=0 should it? Maybe x=0 ...