Here is the question:
This is my work thus far.
Use the Taylor series for the exponential function $e^x$: $$\sum_{k=0}^\infty \frac{\lambda^ke^{sk}}{k!}=\sum_{k=0}^\infty \frac{\left(\lambda e^{s}\right)^k}{k!}=e^{\lambda e^s}. $$
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Use the Taylor series for the exponential function $e^x$: $$\sum_{k=0}^\infty \frac{\lambda^ke^{sk}}{k!}=\sum_{k=0}^\infty \frac{\left(\lambda e^{s}\right)^k}{k!}=e^{\lambda e^s}. $$