Houjun Liu

power series

a power series centered at \(a\) is defined with \(c_{n} \in \mathbb{R}\), whereby:

\begin{equation} f(x) = \sum_{n=0}^{\infty} c_{n}(x-a)^{n} \end{equation}

meaning it is written as \(c_0 + c_1(x-a) + c_2(x-a)^{2} + c_3 (x-a)^{3} + \cdots\)

radius of convergence

  • there is a radius of convergence \(R \geq 0\) for any power series, possibly infinite, by which the series is absolutely convergent where \(|x-a| < R\), and it does not converge when \(|x-a| > R\) , the case where \(|x-a| = R\) is uncertain
  • ratio test: if all coefficients \(c_{n}\) are nonzero, and some \(\lim_{n \to \infty} \left| \frac{c_{n}}{c_{n+1}} \right|\) evaluates to some \(c\) — if \(c\) is positive or \(+\infty\), then that limit is equivalent to the radius of convergence
  • Taylor’s Formula: a power series \(f(x)\) can be differentiated, integrated on the bounds of \((a-R, a+R)\), the derivatives and integrals will have radius of convergence \(R\) and \(c_{n} = \frac{f^{(n)}(a)}{n!}\) to construct the series

linear combinations of power series

When \(\sum_{n=0}^{\infty} a_{n}\) and \(\sum_{n=0}^{\infty} b_{n}\) are both convergent, linear combinations of them can be described in the usual fashion:

\begin{equation} c_1 \sum_{n=0}^{\infty} a_{n}+ c_2 \sum_{n=0}^{\infty} b_{n} = \sum_{n=0}^{\infty} c_1 a_{n} + c_2 b_{n} \end{equation}

some power series

geometric series

\begin{equation} 1 + r + r^{2} + r^{3} + \dots = \sum_{n=0}^{\infty} r^{n} = \frac{1}{1-r} \end{equation}

which converges \(-1 < r < 1\), and diverges otherwise.

exponential series

\begin{equation} 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots = \sum_{n=0}^{\infty} \frac{x^{n}}{n!} = e^{x} \end{equation}

which converges for all \(x \in \mathbb{R}\).

absolutely convergent


\begin{equation} \sum_{n=0}^{\infty} |a_{n}| \end{equation}

converges, then:

\begin{equation} \sum_{n=0}^{\infty} a_{n} \end{equation}

also converges.

This situation is called absolutely convergent.