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Let $$\varphi(x) = e^{-x} + \frac{x}{1!} \cdot e^{-2x} + \frac{3x^{2}}{2!}\cdot e^{-3x} + \frac{4^{2} \cdot x^{3}}{3!} \cdot e^{-4x} + \cdots$$

Then what is the value of: $$ \displaystyle\lim_{t \to 0} \frac{\varphi(1+t) - \varphi(t)}{t}$$

I am not getting any idea as to how to proceed for this problem. I tried summing up the expression but no avail. I also tried differentiating $\varphi(x)$ since the limit quantity which we require seems to involve derivative but again i couldn't find any pattern. Any ideas on how to solve this problem.

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    @chandru1: are you sure the second term is $e^{-3x}$, not $e^{-2x}$?2011-02-02
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    What is the source of the problem?2011-02-02
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    What is the coefficient in the term with $n!$? $(n+1)^{n-1}$? The question is not very clear.2011-02-02
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    I presume the individual terms of $\varphi(x)$ are ${n^{n-1}\cdot x^{n-1}\over n!}\cdot e^{-nx}$ - those terms are virtually identical to the Taylor series for the Lambert function W and given the form (it's essentially $e^{-x}$ times a power series in $xe^{-x}$) I strongly suspect that $\varphi$ has some conveniently explicit form...2011-02-02
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    @All: Question edited now.2011-02-02
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    If we let $z=e^{-x}$ then $x=-ln(z)$ and $\varphi(-ln(z)) = z+\frac{-ln(z)}{1!}\cdot z^2 + \frac{3\cdot ln^2(z)}{2!}\cdot z^3 + \cdots $. This is something like cepstral function on the left side and some Z-transform on the right side.2011-02-02

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