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Proving a certain identity involving the exponential function

Prove the following identity involving the exponential function

$$\frac{1}{1-e^{ -x}}=\sum_{k=1}^\infty \frac{x^k}{k!} $$

where $x\in \mathbb{R} $

Hint: Consider the integral $\int_0^x \frac{1}{1-e^{ -u}}du$

I have some doubts regarding the hint.

\begin{align}

\int_0^x \frac{1}{1-e^{ -u}}du & = \ln \left| \frac{1-e^{ -x}}{1-e^{ -0}} \right|

= \ln \left| 1-e^{ -x}\right|

& \leq \ln \left| e^{ -x}\right| + \ln \left| 1-e^{ -0}\right| \\

& \leq \ln \left| e^{ -x}\right| + \ln 2

\end{align}

Do I need to justify that the value of the last inequality is $x$?

Could someone give me a hand with the proof?

A:

The hint is quite unclear. You should notice that

$$

\int_0^x\frac{u^k}{k!}du=\frac{x^{k+1}}{k+1}.

$$

All in all:

$$

\sum_{k=0}^\infty\frac{x

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