Sequences and Series

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Sequences and Series

• The short-short version ?!

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What is a sequence?

• A sequence can be thought of as an infinite
  list of numbers in order. More formally, as a
  function from the set of positive integers to the
  set of reals.
• Examples
• 1) an 1, 2, 3, . or an n2) an 1,
  or an 1/n3) an
  or an 1/2n

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Find the first 5 terms of the following sequences.

• an n/(n1)
• an
• 2) an (-4)/n!
• an

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What can we do with sequences?

• As sequences are functions, we can
• - graph them,
• - we can find their limit at infinity, (if the
  limit exists and is a real number we say the
  sequence is convergent, otherwise the sequence is
  divergent)
• - we can decide if they are increasing or
  decreasing,
• - we can decide if they are bounded

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What is a series?

• A series or infinite series is simply the sum
  of infinitely many terms
• a1 a2 a3 an
• Short hand or

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What does it mean to add up infinitely many terms?

• Consider an infinite series
• and let sn a1 a2 an
• be the nth partial sum of the series,
• then we say that
• a) if the sequence s is convergent and the
  limit as n approaches infinity exists as a real
  number, say s, then the series converges and we
  write
• a1 a2 an s or
• b) otherwise, the series diverges.

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The Geometric Series

• A geometric series is of the form
• a ar ar2 ar3 arn-1
• It converges if r lt 1 and
• its sum is a/(1-r). Otherwise, the geometric
  series is divergent.

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Examples

• Find the sum of the geometric series
• a) 1 0.4 0.16 0.064
• b) 2 2/3 2/9 2/27 2/81
• c) 0.1111111

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Power Series

• A power series is of the form
• c0 c1 x c2 x2 c3 x3
• where the cs are the coefficients and the x
  is a variable.

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Example

• If we let the constants be all 1 in the power
  series, then we get the geometric series
• 1 x x2 x3 1/(1-x)
• if x lt 1, which is the radius of
  convergence of this power series.

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Note

• We can use power series to represent certain
  functions, with radius of convergence, which
  allows us to both differentiate and integrate
  functions using this form term by term.
• Question becomes which functions are
  representable?

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Taylor Series

• If a function f has a power series expansion
  at some number a, then it has the form
• f(x)
• this is called the Taylor series of the
  function f at a (or centered at a).

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Maclaurin Series

• The special case of the Taylor series when a
  0 is simply of the form
• f(x)
• Lets look at some examples!