Deriving Series for ln(1+x)

Real Analysis · Axiom Academy

EXAMPLE Deriving the Series for Integrate the geometric series term by term, then pin down exactly where the result is valid. Find a power series for by integrating the geometric series for term by term, and determine the full interval on which the series equals . Partial sums converging to the curve The partial sums hug ever more closely on . Each added term tightens the fit; outside that window the polynomials peel away from the curve. You derived a power series for straight from the geometric series — here is what made it work. Integrate a known series: integrates term by term to give . The radius is inherited: term-by-term integration leaves R=1 unchanged from the geometric series. Endpoints need their own test: at x=1 the alternating harmonic series converges (to ); at x=-1 it becomes the negative harmonic series and diverges. Interval of validity: — the right endpoint is included, the left is not. The same move — start from a geometric series and integrate or differentiate term by term — produces the series for , and underlies the expansions of many other functions.

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