Summing up via Taylor series: Methods survey

This strategy works as follows. Given a series of real numbers, find a power series and a number x so that when this number is put into the power series, we get the given series. If we know the sum of that power series (which can be sometimes done), then we consequently also sum up the given series. Example shows it best.

Example: We want to sum up the series  .

We want to guess a preferably simple power series that can yield our series. Since a standard power series has the power to k in the numerator, the given series has to be first rearranged a bit and then we can make a good guess:

Indeed, if we substitute x = 1/2 into the chosen power series, we do get the given series. What do we know about the power series in question? We investigated it in this problem in Solved Problems - Series of functions and found that

Our particular x = 1/2 fits in the region of convergence of this power series, thus we can substitute it into the above equality and conclude that

For more examples see Summing up series in Theory - Introduction and this problem and this problem in Solved Problems - Summing up series.

Approximating series
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