SERIES AND CONVERGENCE

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Chapter 9.2

• SERIES AND CONVERGENCE

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After you finish your HOMEWORK you will be able
to

• Understand the definition of a convergent
  infinite series
• Use properties of infinite geometric series
• Use the nth-Term Test for Divergence of an
  infinite series

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INFINITESERIES

• An infinite series (aka series) is the sum of the
  terms of an infinite sequence.
• Each of the numbers, , are called terms of
  the series.

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CONVERGENT AND DIVERGENT SERIES

• For the infinite series , the n-th partial
  sum is given by .
• If the sequence of partial sums, , converges
  
• to , then the series converges. The
  limit is called the sum of the series.
• If diverges, then the series diverges.
• Series may also start with n 0.

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THE BATHTUB ANALOGY
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DIVERGE VERSUS CONVERGE

• Consider the series
• What happens if you continue adding 1 cup of
  water?

• Consider the series
• How is this situation different?
• Will the tub fill?

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TELESCOPING SERIES

• What do you notice about the following series?
• What is the nth partial sum?

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CONVERGENCE OF A TELESCOPING SERIES

• A telescoping series will converge if and only
  if approaches a finite number as n approaches
  infinity.
• If it does converge, its sum is

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GEOMETRIC SERIES

• The following series is a geometric series with
  ratio r.

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THEOREM 9.6CONVERGENCE OF A GEOMETRIC SERIES

• A geometric series with ratio r diverges if
  .
• If then the series converges
  to

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THEOREM 9.7PROPERTIES OF INFINITE SERIES

• If is a real
  number,
• then the following series converge to the
• indicated sums.

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THEOREM 9.8LIMIT OF nth TERM OF A CONVERGENT
SERIES

• If converges, then
• Why?

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The nth-Term TestTHEOREM 9.9

• If , the
• infinite series
• diverges.