POWER SERIES REVIEW

Assume that a power series is given in the form

$$f(x) = \sum_{n=0}^\infty a_n(x-x_0)^n .$$

Convergence

• A series converges at a point x if
  
  $$\lim_{m\to\infty}\sum_{n=0}^m a_n(x-x_0)^n$$
  
  
  exists.
• A series converges absolutely at x if
  
  $$\sum_{n=1}^\infty \vert a_n(x-x_0)^n\vert$$
  
  
  converges.
• The ratio test: suppose $\vert x-x_0\vert\lim\limits_{n\to\infty}\vert\frac{a_{n+1}}{a_n}\vert = L .$
  a) If L<1, the series converges absolutely.
  b) If L>1, the series diverges.
• Alternating series: if $0\leq -(x-x_0)\frac{a_{n+1}}{a_n} < 1$,
  then the series converges.
• If f(x) converges at x₁, then
  f(x) converges absolutely for |x-x₀| < |x₁-x₀|.
• If f(x) diverges at x₁, then
  f(x) diverges for |x-x₀| > |x₁-x₀|.
• Radius of convergence $\rho$:
  f(x) converges absolutely for $\vert x-x_0\vert< \rho$ and
  f(x) diverges for $\vert x-x_0\vert> \rho$.
  The interval of convergence is $(x_0-\rho,x_0+\rho)$.

MORE POWER SERIES REVIEW

Algebra of Series:

let $g(x) = \sum\limits_{n=0}^\infty b_n(x-x_0)^n .$

• If f and g converge at x then
  
  $$f(x)\pm g(x) = \sum_{n=0}^\infty (a_n\pm b_n)(x-x_0)^n$$
  
  
  converges at x.
• If f and g converge at x then
  
  $$\begin{eqnarray*}f(x)g(x) & = & \big(\sum_{n=0}^\infty a_n(x-x_0)^n\big) \big(\s... ...infty b_n(x-x_0)^n\big)\\ & = &\sum_{n=0}^\infty c_n(x-x_0)^n , \end{eqnarray*}$$
  
  with $c_n = a_0b_n + a_1b_{n-1} + \cdots + a_nb_0$, converges at x.
• If f and g converge at x then
  
  $$\frac{f(x)}{g(x)} = \sum_{n=0}^\infty d_n(x-x_0)^n,$$
  
  
  with $a_n = d_0b_n + d_1b_{n-1} + \cdots + d_nb_0$.
• If f and all derivatives exist for $\vert x-x_0\vert< \rho$, derivative series can be computed using termwise differentiation:
  $f'(x) = \sum\limits_{n=1}^\infty na_n(x-x_0)^{n-1}$, and
  $f''(x) = \sum\limits_{n=2}^\infty n(n-1)a_n(x-x_0)^{n-2}$.
• Shift of index:
  $\sum\limits_{n=0}^\infty a_n(x-x_0)^n = \sum\limits_{m=k}^\infty a_{m-k}(x-x_0)^{m-k}$ .

Taylor series

: $f(x) = \sum\limits_{n=0}^\infty\frac{f^{n}(x_0)}{n!}(x-x_0)^n$.