Infinite Series

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Lecture 15

• Infinite Series

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Lecture 15 Objectives

• Find the partial sums of
• Geometric Series
• Telescoping Series
• Determine the convergence (and find the sum) or
  divergence of
• Geometric Series (or linear combinations of
  these)
• Telescoping Series
• Determine the divergence of Series using the
  nth-Term (Divergence) Test.

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Example

• Consider the following infinite series (sum) of
  real numbers

• Question What is this infinite sum?

• Answer 2

• Reason The infinite sum is the limit of the
  (partial) sum of the first n terms as n ? ?.

• Caution This series is not the same as the
  sequence

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Picture
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Calculation of Partial Sums
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Notation

• The infinite sum denotes the limit of partial
  sums. I.e.

Or using the Sigma notation
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Geometric Series

• This is a series of the form a ar ar2
  ar3
• I.e. the ratio between successive terms is a
  constant r
• The nth partial sum can be found by
• sn a ar ar2 arn?1
• a(1 ? rn)/(1 ? r) (if r ? 1)
• Note When r lt 1, rn ? 0, so sn ? a/(1 ? r)
• When r 1, sn na, so sn ? ?? (if a ? 0)
  
• Otherwise, rn diverges, so sn diverges.

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Geometric Series
Thus,
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Example For the geometric series

• Find the nth partial sum.
• Is this series convergent?
• If yes, find its sum.

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Example For the geometric series

• Find the nth partial sum.
• Is this series convergent?

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Example Express the repeating decimal
911.911911as a ratio of two integers.
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Rules for Convergent Series
Example Find the sum of the series
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Example (Telescoping Series) For the series

• Find the nth partial sum.
• Is this series convergent?
• If yes, find its sum.

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Example (Telescoping Series) For the series

• Find the nth partial sum.
• Hint Use partial fraction decomposition.
• Is this series convergent?
• If yes, find its sum.

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Example Check the divergence of the series 1
2 1 2 1 2 Or 1 ? 1 1 ? 1 1 ?
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• Note If the terms we keep adding do not tend to
  0 in the limit, then the infinite sum must
  diverge.

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In general
In other words

• Caution
• If limn an 0, then ?n an may or may not
  converge.

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Example Show that the seriesis divergent.
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Example

• Consider the following so-called Harmonic
  series

• Question Is this series convergent?

• Answer No, but this is not obvious.

• Caution The term sequence actually converges to
  0.

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Why is the harmonic series divergent?Reason

• The Harmonic series

can be shown (see Picture) to represent an area
that is ? the integral
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Picture
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The Integral Test
Reason
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Picture
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Example

• Show that the p-series
• converges if p gt 1,

and diverges if p ? 1.
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Example

• Which of the following series is convergent and
  which is divergent?

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• Thank you for listening.
• Wafik