Glossary

Convergence

Criticality: 3

The property of an infinite series where its sequence of partial sums approaches a finite limit. For power series, this defines the specific interval or radius of x-values for which the series accurately represents the function.

Example:

When working with a geometric series like $\sum_{n=0}^{\infty} x^n$ , we know it only achieves convergence when the absolute value of x is less than 1, meaning it sums to a finite value only within that range.

Derivative of a Power Series

Criticality: 3

The process of finding the derivative of a function represented by a power series by differentiating each term of the series individually, which results in a new power series.

Example:

If you have the power series for $\sin(x)$ , taking the derivative of a power series term by term will yield the power series for $\cos(x)$ , demonstrating a powerful connection between functions and their series representations.

General Term

Criticality: 3

The formula, often involving 'n', that defines the nth term of an infinite series, allowing the series to be written concisely using summation notation.

Example:

For the power series of $\sin(x)$ , the general term is $\frac{(-1)^n x^{2n+1}}{(2n+1)!}$ , which helps us generate any term in the series by plugging in different values for n.

Power Series

Criticality: 3

An infinite series representation of a function, expressed as a sum of polynomial terms involving powers of (x-r), where r is the center of the series.

Example:

The power series for $e^x$ is $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots$ , which allows us to approximate $e^x$ using a polynomial for any given x value. This is a fundamental example of a Power Series.