Power Series

A power series is a type of infinite series that can be thought of as a polynomial with infinitely many terms. It is a function of $x$ defined by:

${\sum }_{n=0}^{\infty }{c}_n(x-a{)}^n=c_0+c_1(x-a)+c_2(x-a{)}^2+\dots$

Where:

• $c_n$ are the coefficients.
• $a$ is the center of the series.

Radius and Interval of Convergence

Unlike a finite polynomial, a power series may only converge for certain values of $x$.

1. Radius of Convergence ($R$): There exists a number $R\ge 0$ such that the series converges if $|x-a|<R$ and diverges if $|x-a|>R$.
2. Interval of Convergence: The set of all $x$ for which the series converges. This interval is always centered at $a$ and has a width of $2R$. Note that convergence at the endpoints ($x=a-R$ and $x=a+R$) must be checked separately.

Finding the Radius of Convergence

The most common method to find $R$ is using the Ratio Test:

$L={\lim }_{n\to \infty }\left|\frac{c_{n+1}(x-a{)}^{n+1}}{c_n(x-a{)}^n}\right|=|x-a|{\lim }_{n\to \infty }\left|\frac{c_{n+1}}{c_n}\right|$

The series converges when $L<1$.

Relationship to Taylor Series

Every Taylor series is a power series. However, not every power series is a Taylor series for a specific function (though most “useful” ones are).

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Draft note: Add section on term-by-term differentiation and integration.