Concept preview

What you will be able to do

1. 01Start from the geometric series $\dfrac{1}{1-x}=\sum_{n=0}^{\infty}x^{n}$ (valid for $|x|<1$) and generate new power-series representations by SUBSTITUTING an expression in place of $x$.
2. 02Build further series by DIFFERENTIATING or INTEGRATING a known series term by term, keeping every sign and index shift correct.
3. 03State the interval of convergence of each new series, adjusting the endpoints correctly when a substitution rescales the variable, and multiply a series by a power of $x$ to represent related functions.

Everything in this lesson grows from a single seed you already met: the geometric series $\dfrac{1}{1-x}=\sum_{n=0}^{\infty}x^{n}=1+x+x^{2}+x^{3}+\cdots,\qquad |x|<1.$ The right side is a power series centred at $0$, and the equality holds only on the open interval $(-1,\,1)$ — its interval of convergence. Outside that interval the terms do not shrink to zero and the sum runs off to infinity, so the identity is meaningless there. Keep both halves of the statement in mind at all times: the formula AND the interval. A power-series representation is only as good as the set of inputs on which it is true.