Problem: Evaluate where is a real number with .
Solution: Looking at that all too common denominator, we do a partial fraction decomposition in hopes of telescoping series. The summation becomes
Common Taylor series knowledge tells us that
which convenient fits the first part of the summation. As for the second part, we get
from the same Taylor series. Combining the results, our answer is then
Comment: Even though the trick at the beginning didn’t actually get much to telescope, the idea certainly made it easier to recognize the Taylor series. Algebraic manipulations are nifty to carry around and can be applied in problems wherever you go.
Practice Problem: Show that .
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