Calculus II - 27
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Calculus II - 27

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Stewart Calculus 11.9

Stewart Calculus 11.9

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Calculus II - 27 Calculus II - 27 Presentation Transcript

  • 11.9 Representations offunctions as power series We already know that = + + + + ··· = converges to when < < . On the contrary, can be expressed a power series = + + + + ··· = when < < .
  • Ex: Find a power series representation for + and the radius of convergence.
  • Ex: Find a power series representation for + and the radius of convergence.When | |< , = + + + ··· = = View slide
  • Ex: Find a power series representation for + and the radius of convergence.When | |< , = + + + ··· = =Therefore, = ( ) = ( ) + = = View slide
  • Ex: Find a power series representation for + and the radius of convergence.When | |< , = + + + ··· = =Therefore, = ( ) = ( ) + = =It converges when | |< , i.e. < < .
  • Ex: Find a power series representation for + and the radius of convergence.When | |< , = + + + ··· = =Therefore, = ( ) = ( ) + = =It converges when | |< , i.e. < < .The radius of convergence is .
  • Ex: Find a power series representation for and the radius of convergence. +
  • Ex: Find a power series representation for and the radius of convergence. +When | |< , = + + + ··· = =
  • Ex: Find a power series representation for and the radius of convergence. +When | |< , = + + + ··· = =Therefore, ( ) = = = + + + = =
  • Ex: Find a power series representation for and the radius of convergence. +When | |< , = + + + ··· = =Therefore, ( ) = = = + + + = =It converges when | / |< , i.e. < < .
  • Ex: Find a power series representation for and the radius of convergence. +When | |< , = + + + ··· = =Therefore, ( ) = = = + + + = =It converges when | / |< , i.e. < < .The radius of convergence is .
  • Ex: Find a power series representation for and the radius of convergence. +
  • Ex: Find a power series representation for and the radius of convergence. + ( ) = = = + + + = =
  • Ex: Find a power series representation for and the radius of convergence. + ( ) = = = + + + = =It converges when | / |< , i.e. < < .
  • Ex: Find a power series representation for and the radius of convergence. + ( ) = = = + + + = =It converges when | / |< , i.e. < < .The radius of convergence is .
  • Ex: Find a power series representation for and the radius of convergence. + ( ) = = = + + + = =Therefore, ( ) ( ) + = + = + + = =It converges when | / |< , i.e. < < .The radius of convergence is .
  • Ex: Find a power series representation for ( ) and the radius of convergence.
  • Ex: Find a power series representation for ( ) and the radius of convergence. ( )=
  • Ex: Find a power series representation for ( ) and the radius of convergence. ( )= = + + + + ···
  • Ex: Find a power series representation for ( ) and the radius of convergence. ( )= = + + + + ··· = ··· +
  • Ex: Find a power series representation for ( ) and the radius of convergence. ( )= = + + + + ··· = ··· + = + =
  • Ex: Find a power series representation for ( ) and the radius of convergence. ( )= = + + + + ··· = ··· + = + = Take = , we get = .
  • Ex: Find a power series representation for ( ) and the radius of convergence. ( )= = + + + + ··· = ··· + = + = Take = , we get = . The radius of convergence is .
  • Theorem (term-by-term diff. and int.):If the power series ( − ) has a radiusof convergence , then ∞ ( )= = ( − )is differentiable on the interval ( − , + )and ∞ − ( )= = ( − ) + ∞ ( − ) ( ) = + = +The radii of both series are .
  • Ex: Find a power series representation for ( ) and the radius of convergence.
  • Ex: Find a power series representation for ( ) and the radius of convergence. = ( )
  • Ex: Find a power series representation for ( ) and the radius of convergence. = ( ) =( + + + + ···)
  • Ex: Find a power series representation for ( ) and the radius of convergence. = ( ) =( + + + + ···) = + + + ···
  • Ex: Find a power series representation for ( ) and the radius of convergence. = ( ) =( + + + + ···) = + + + ··· = ( + ) =
  • Ex: Find a power series representation for ( ) and the radius of convergence. = ( ) =( + + + + ···) = + + + ··· = ( + ) = The radius of convergence is .