Interval of Convergence

Domain

• Is the set of x values for which the power series converges
• Every power series converges at c, meaning c always lies on the domain of f
• May be one of 3 options
  • A single point (c)
  • An interval centered at c
  • All real numbers

Radius

• When the domain is an interval centered at c
• There is a real number R, called the radius of convergence
  • If $R=0$, it only converges at $c$
  • If $R=\infty$, it converges for all real numbers
• If $R>0$, it…
  • Converges for $|x-c|<R$
  • Diverges for $|x-c|>R$

Interval of Convergence

• The range that the series converges is called the interval of convergence
• Finding the interval of convergence
  1. Find the interval
     • Use the ratio test to find the range of values $x$ which satisfy ${\lim }_{x\to \infty }|\frac{a_{n+1}}{a_n}|<1$
     • If a Geometric Series, you can instead find when $|r|<1$ (because that is the condition for convergence in the GST)
  2. Check the endpoints (plug in for x) using any other convergence test to see if they are included

Calculus Properties

• If a power series has a $R>0$, it is differentiable on $(c-R,c+R)$
  • $f^{\prime }(x)={\sum }_{n=1}^{\infty }n{a}_n(x-c{)}^{n-1}$
  • $\int f(x)dx=C+{\sum }_{n=0}^{\infty }\frac{a_n}{n+1}(x-c{)}^{n+1}$
• The interval of convergence is the same for a function’s derivative and anti-derivative, though it may differ in including the endpoints