7 Power Series and Taylor Series Free Online FlashCards

What is a power series centered at $a$?

A power series centered at $a$ has the form $\sum_{n=0}^{\infty} c_n(x-a)^n$, where the $c_n$ are constants.

When does the geometric series converge?

The geometric series satisfies $\sum_{n=0}^{\infty}r^n=\frac{1}{1-r}$ when $|r|<1$.

What is the radius of convergence?

The radius of convergence is the number $R$ such that a series centered at $a$ converges for $|x-a|<R$ and diverges for $|x-a|>R$.

What is the interval of convergence?

The interval of convergence is the complete set of real $x$-values for which the power series converges, including any endpoints that pass their tests.

How should endpoints be handled after finding $R$?

For a power series, the ratio test usually gives $|x-a|<R$. The boundary values $x=a-R$ and $x=a+R$ must then be tested separately.

What are the radius and interval for $\sum_{n=1}^{\infty}\frac{(x-2)^n}{n3^n}$?

For $\sum_{n=1}^{\infty}\frac{(x-2)^n}{n3^n}$, the radius is $R=3$ and the interval of convergence is $[-1,5)$.

How is a power series differentiated within its radius?

Inside $|x-a|<R$, differentiation is term-by-term: $\frac{d}{dx}\sum_{n=0}^{\infty}c_n(x-a)^n=\sum_{n=1}^{\infty}nc_n(x-a)^{n-1}$.

How is a power series integrated term by term?

Inside the interval of convergence, integrate term-by-term: $\int\sum_{n=0}^{\infty}c_n(x-a)^n dx=C+\sum_{n=0}^{\infty}\frac{c_n}{n+1}(x-a)^{n+1}$.

What is the Taylor series of $f$ about $a$?

The Taylor series of $f$ about $a$ is $\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$.

What is a Maclaurin series?

A Maclaurin series is a Taylor series centered at $a=0$: $\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$.

What is the Maclaurin series for $e^x$?

The Maclaurin series for $e^x$ is $\sum_{n=0}^{\infty}\frac{x^n}{n!}$, and its radius of convergence is $R=\infty$.

What is the convergence interval for the series of $\ln(1+x)$?

The series $\ln(1+x)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^n}{n}$ converges on $-1<x\le 1$.