MATH1012 - Lecture 22

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MATH1012 - Lecture 21
Skeleton Slides

Convergence

• $R=\underset{n\to \infty }{\lim }\left|\frac{b_n}{b_{n+1}}\right|$ is called the radius of convergence of the power series
  • Read last lecture: Applying the Ratio Test
• Three possible behaviours:
  • Absolutely convergent when $x=c$,
    • Divergent otherwise
  • Absolutely convergent for all $x$
  • If $R$ is a positive real number:
    • Absolutely convergent if $|x-c|<R$
    • Divergent if $|x-c|>R$
    • Has to be determined case by case if $|x-c|=R$
• Proof in Unit Reader

Conditional Convergence

To converge conditionally, you get a series where +ve terms cancel out -ve terms resulting in the series converging. However, if the sign of either the +ve or -ve terms changed, the series would diverge. If you can change the sign of the numbers in the series and it still converges, it converges absolutely.

• When $x$ lies in the interval of convergence, the power series behaves exactly like a giant polynomial
  • The interval of convergence is $c-R\le x\le c+R$ where $0<R<\infty$
• Thus we can describe where a power series converges by:
  • Finding its centre $(c)$
  • Solving the radius of convergence $R$
  • Checking $x=c-R,x=c+R$
• Make sure you check what happens at the boundaries of the interval
  • Series may converge absolutely, converge conditionally or diverge

Differentiation

• For a power series with radius of convergence $R>0$:
  • The series can be differentiated term by term
  • The radius of convergence of the derived series is also R
• If $f(x)={\sum }_{n=0}^{\infty }{b}_n(x-c{)}^n$, then $f^{\prime }(x)={\sum }_{n=1}^{\infty }n{b}_n(x-c{)}^{n-1}$
  • $f^{\prime \prime }(x)={\sum }_{n=2}^{\infty }n(n-1)b_n(x-c{)}^{n-2}$

Taylor Polynomial

• See Taylor Polynomials

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MATH1012 - Lecture 23