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However, is the same true if it is multiplied by say t, t^2, t^3,t^n.

i.e. t*exp(-t) for example.

the exp is decaying to zero faaster than t is, so it goes to zero in the limit. But there are functions that decay to zero but their integral is not finite because the rate of decay is not *fast enough*.

Would the integral of exp multiplied by any power of t ALWAYs converge to a finite number?