Problem: Evaluate .
Solution: Let . Then , a well-known Taylor series. So we want to integrate this:
by parts using and . Substituting in the last integral, we have
So . Thus
for some constant . Using our knowledge that , , and by L’Hopital twice, we see that
Then setting we obtain and . QED.
Comment: This was a pretty tough problem that required you to compound a lot of calculus knowledge all into a single problem – series, integration by parts, limits. Recognizing all the steps was the first part; following through with the right computations was another. Still, there weren’t really any super clever tricks, mostly just standard substitutions and approaches applied in a somewhat non-standard way. Makes for a very nice problem.
Practice Problem #1: Show that .
Practice Problem #2: Evaluate . Can you also find ?
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