Problem: (Problem-Solving Through Problems – 5.4.15) Let and for . Show that
Solution: Well, the LHS looks suspiciously like a Taylor series, so maybe we can find a function. A good choice is . Notice that . Testing a few derivatives, we hypothesize that
By induction, we easily have
which exactly satisfies
so we indeed have . Then and the Taylor series of centered at zero is
Hence our desired sum is the above series evaluated at , which is simply
Comment: A neat problem resulting from Taylor series. They are useful in all sorts of ways, especially evaluating other series. If we can reduce a given series to the Taylor series of a function like we did in this problem, evaluating it becomes plugging in a point.
Practice Problem: (Problem-Solving Through Problems – 5.4.17) Show that the functional equation
is satisfied by
Leave a Reply
You must be logged in to post a comment.