MATH1012 - Lecture 21

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MATH1012 - Lecture 20
Skeleton Slides (Wk7)
Skeleton Slides (Wk8)

Absolute and Conditional Convergence

• The series ${\sum }_{n=1}^{\infty }{a}_n$ is absolutely convergent if ${\sum }_{n=1}^{\infty }|a_n|$ is convergent
• If ${\sum }_{n=1}^{\infty }|a_n|$ is not convergent, the series is conditionally convergent

The Ratio Test

• The ratio test is probably the most useful test we have discussed
  • You do not need to be able to integrate the sequence
  • The test works for many cases and varying forms of series
    • But this is not always the case! i.e. when $L=1$
• Suppose $a_n$ is a sequence such that ${\lim }_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|=L$
• We are given three cases:
  • $L<1$ means that ${\sum }_{n=1}^{\infty }{a}_n$ is absolutely convergent
  • $L=1$ gives no information
  • $L>1$ means that ${\sum }_{n=1}^{\infty }{a}_n$ is divergent

The sum of some more complicated looking series can be solved by splitting the series into parts that look familiar. An example is presented in the lecture recording utilising the formula for a geometric series.

Power Series

• A power series is a series of the form: ${\sum }_{n=0}^{\infty }{b}_n(x-c{)}^n$
  • $x$ is a variable
  • $b_n$ is a sequence of coefficients
  • $c$ is some fixed number, called the centre of the power series
• This is a function of $x$ for all the values of $x$ where the series is convergent
  • Full expanded general formula for power series on slides
  • This is important in the calculation of function values by computers and calculators

Applying the Ratio Test

• Let $a_n=b_n(x-c{)}^n$
  • $L=\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|$
• Let $R=\underset{n\to \infty }{\lim }\left|\frac{b_{n+1}}{b_n}\right|⟹L=\frac{|x-c|}{R}$
• We are given three cases:
  • $R=0$ thus $L$ is undefined
  • $R$ does not exist (i.e. limit goes to infinity) → $L=0$
  • $R$ is a positive real number

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MATH1012 - Lecture 22
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