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We have the following series involving some Laplace transformed function $F(s)$,

$$sum^{infty}_{n=0} left( frac{s}{a} right)^{n} F(s).$$

Using the following formula,

$$left( frac{1}{a} right)^{n} mathcal{L} { sum^{infty}_{n=0} f^{(n)}(t) } = s^{n}F(s) + left( frac{1}{a} right)^{n} left( sum^{n-1}_{m=0} f^{(m)}(0) s^{(n-1)-m}right) ,$$

if $a rightarrow infty$ we truncate the series and take the following leading terms,

$$mathcal{L}^{-1}{F(s)+frac{s}{a}F(s)}=f(t)+frac{1}{a}f^{prime}(t)+frac{1}{a}delta(t)f(0).$$

Then we must make assumptions regarding $s$ and its relationship to $a$ which prevents us from using this simplified formula when $t rightarrow 0$. Yet if we inverse Laplace transform the entire series and take the leading term afterwards then we are no longer making these assumptions. Instead this depends on the value of the function, $f$, at $t=0$, its derivative, and the Dirac Delta function. Are these equivalent or is there a discrepancy between these two truncations? Are neither, one, or both of these descriptions suitable when describing $t rightarrow 0$?