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MAT136H1 Lecture Notes - Antiderivative

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11.10 Infinite Sequences & Series
Taylor & Maclaurin Series
Question #4 (Medium): Evaluating Indefinite Integral as Infinite Series
Any given function inside the indefinite integral can be expressed as a infinite series. Then put that sum
expression inside the indefinite integral, then focus on the power of , since the integral is with respect
to variable . Then all else is treated as number, including the series counter , but power is raised by
, and that power needs to be offset by the reciprocal fraction, thus the outcome being similar series
expression with power raised to one more, and offsetting coefficient added to the series expression.
Remember to add the arbitrary constant factor of since it is indefinite integral.
Sample Question
Evaluate the indefinite integral as an infinite series.
First take the expression inside the indefinite integral and express as an infinite series.
 
  
Then to evaluate the indefinite integral, put this summation expression inside the integral:
 
   
 . Remember add the arbitrary constant
factor of !
The radius of convergence is:
and  
, then 
. Therefore the radius of
convergence is
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