Relation of laplace transform with power series

[and_5]

Hi, just wonder if anyone can help

Homework Statement

Apparently there is a relation between laplace transform and power series. http://www.jstor.org/stable/pdfplus/2305640.pdf?acceptTC=true states that if the discrete variable n of a power series is replaced by a continuous variable lambda, the infinite sum becomes an integral with boundaries between zero and infinity. But where does the d'lambda' of the integrand come from? This looks a bit like the Riemann sum, but if I haven't understood it wrong, the dx in the Riemann integrand is supposed to be a replacement for the discrete delta x in the Riemann sum and thus the operation is said to be integrate with respect to x? There isn't any delta lambda in the original power series which the laplace transform is made analogous to?

Second, I don't really know why one can arbitrarily plug in any formula into the laplace transform integral and get the transform of it. The whole procedure is just like evaluating a power series with the function of interest acting as the accompanying coefficients of a continuous power series infinite sum of [f(t)x^t], isn't that quite arbitrary? or did this manipulation just happen to turn out to be something quite useful in solving differential equations?

Thanks.

Homework Equations

The Attempt at a Solution

 

[lippyka]

The d'lambda' of the integrand comes from the differential of the continuous variable lambda. This is analogous to the delta x in Riemann sum, which corresponds to the differential of the discrete variable x. The formula for the Laplace Transform can be plugged into the integral because it is the integral representation of the infinite sum of a continuous power series. It is useful for solving differential equations because it can be used to transform the equation into a simpler form.