Power Series: Convergence, Operations, Analyticity, and Generating Functions

All flashcards

Cauchy–Hadamard formula

For $\sum_{n=0}^{\infty}a_n(x-c)^n$, the radius is $r=1/\limsup_{n\to\infty}|a_n|^{1/n}$.

Cauchy product

The coefficient of degree $n$ in a product is $m_n=\sum_{i=0}^{n}a_i b_{n-i}$.

Formal power series

A coefficient sequence treated algebraically without requiring numerical convergence.

Analytic function

A function that locally equals a convergent power series near every point in its domain.

Analytic continuation

An analytic function on a larger set that agrees with a power series where it converges.

Quiz questions

1. For $\sum_{n=0}^{\infty}a_n(x-c)^n$, suppose $\lim_{n\to\infty}|a_{n+1}/a_n|=L$ for some positive finite $L$. What is its radius of convergence?Answer: $1/L$
2. A power series has radius of convergence $r$. What can be concluded from the radius alone about points where $|x-c|=r$?Answer: Each boundary point must be tested separately.
3. If two power series have coefficients $a_n$ and $b_n$, what is the coefficient of $(x-c)^n$ in their product?Answer: $\sum_{i=0}^{n}a_i b_{n-i}$
4. Within the convergence disc, what happens to a power series' radius when it is differentiated term by term?Answer: The radius remains unchanged.
5. Two analytic functions on a connected open domain have identical derivatives of every order at one point. What follows?Answer: They agree throughout the domain.