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Discover the Path to Mathematical Mastery
An Education That Counts
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Discover the Path to Mathematical Mastery
An Education That Counts
*Be sure to memorize the power series representation for the exponential function \(\texttt{e}^x\), sine \(\texttt{sin}(x)\), cosine \(\texttt{cos}(x)\), the natural logarithm \(\texttt{ln}(x)\), arctangent \(\texttt{arctan}(x)\), as well as the Geometric Series \(\textstyle\frac{1}{1-x}\) provided that. Also, be sure to note the radius and interval of convergence for each power series.
The interval of convergence for a given power series \(f(x)=\sum a_n(x-a)\) is an interval containing every value for \(x\) so that the power series is guaranteed to converge. The interval of convergence may, or may not, include the endpoint. If an endpoint is included a square bracket is used to denote this fact: \((a,b)\), \((a,b]\), \([a,b)\), \([a,b]\).
There are three primary means for determining the interval of convergence.
The radius of convergence corresponding to an interval of convergence can be found taking half of the distance of the interval \((a,b)\)
\(\displaystyle R=\frac{b-a}{2} \)Check with your tutor
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