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Glossary

A

Alternating Harmonic Series

Criticality: 2

The convergent series $\sum^{\infty}_{n=1}\frac{(-1)^{n+1}}{n}$ (or $\sum^{\infty}_{n=1}\frac{(-1)^{n}}{n}$), which is the harmonic series with alternating signs.

Example:

If testing an endpoint of a power series leads to the alternating harmonic series, you know that endpoint is included in the interval of convergence due to conditional convergence.

C

Center of the Interval of Convergence

Criticality: 2

The point $r$ around which a power series is centered, which also serves as the midpoint of its interval of convergence.

Example:

For the power series ∑n=0∞(x−5)nn\sum_{n=0}^{\infty} \frac{(x-5)^n}{n}, the center of the interval of convergence is x=5x=5.

E

Endpoints (of the interval)

Criticality: 3

The two extreme values of $x$ that define the boundaries of the open interval of convergence, which must be individually tested for convergence.

Example:

When finding the interval of convergence for ∑xnn\sum \frac{x^n}{n}, after finding the radius R=1R=1, you must test the endpoints x=−1x=-1 and x=1x=1 separately.

H

Harmonic Series

Criticality: 2

The divergent series $\sum^{\infty}_{n=1}\frac{1}{n}$, where each term is the reciprocal of a positive integer.

Example:

When testing the right endpoint of a power series, if it results in the harmonic series, you know that endpoint is not included in the interval of convergence.

I

Interval of Convergence

Criticality: 3

The set of all $x$-values for which a power series converges, including the endpoints if the series converges at those points.

Example:

For a power series, if the interval of convergence is [−1,1)[-1, 1), it means the series converges for x=−1x=-1 but diverges at x=1x=1.

P

Power Series

Criticality: 3

A series of the form $\sum^{\infty}_{n=0}a_n(x-r)^n$, where $n$ is a non-negative integer, $a_n$ is a sequence of real numbers, and $r$ is a real number representing the center.

Example:

The Maclaurin series for exe^x, ∑n=0∞xnn!\sum_{n=0}^{\infty} \frac{x^n}{n!}, is a power series centered at x=0x=0 that converges for all real numbers.

R

Radius of Convergence

Criticality: 3

A non-negative number, $R$, such that a power series converges for $|x-r| < R$ and diverges for $|x-r| > R$.

Example:

If a power series for sin⁡(x)\sin(x) has a radius of convergence of ∞\infty, it means the series provides an accurate approximation for all real numbers xx.

Ratio Test

Criticality: 3

A test used to determine the convergence or divergence of a series $\sum a_n$ by evaluating the limit $L=|\lim\limits_{n→ \infty}\frac{a_{n+1}}{a_n}|$.

Example:

To find the radius of convergence for ∑xnn!\sum \frac{x^n}{n!}, you'd apply the Ratio Test and find L=0L=0, indicating convergence for all xx.