Representing Functions as Power Series

Every analytic function is locally equal to its Taylor series, but we don't always need the full Taylor machine to find the series. Starting from the geometric series $1/(1-x) = \sum_{n=0}^\infty x^n$, we can build power series for $\ln(1+x)$, $\arctan x$, and many other functions purely by algebraic substitution, differentiation, or integration — without ever computing a derivative. In quantitative finance, the power series for $\ln(1+r)$ governs log-return approximations (log-return $\approx r - r^2/2 + \cdots$), and the series for $\arctan x$ evaluates to $\pi/4$ via Leibniz's formula — a number that appears in the angle parameterisation of risk-neutral densities.

• derive power series by substituting into the geometric series;
• differentiate and integrate power series to obtain new series;
• find power series for $\ln(1+x)$, $1/(1+x^2)$, and $\arctan x$;
• apply these to compute definite integrals as infinite series.

1. The geometric series as the base
2. Power series for $\ln(1+x)$
3. Power series for $\arctan x$
4. Evaluating integrals as power series