Power Series Representations of Functions
Key Questions
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Since
#e^x=sum_{n=0}^infty{x^n}/{n!}# ,by replacing
#x# by#x^2# ,#e^{x^2}=sum_{n=0}^infty{(x^2)^n}/{n!}=sum_{n=0}^infty{x^{2n}}/{n!}# .I hope that this was helpful.
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Recall:
#1/{1-x}=sum_{n=0}^infty x^n=1+sum_{n=1}^infty x^n#
By multiplying by#x# ,
#x/{1-x}=sum_{n=0}^infty x^{n+1}=sum_{n=1}^infty x^n# So,
#{1+x}/{1-x}=1/{1-x}+x/{1-x}=1+sum_{n=1}^infty x^n+sum_{n=1}^infty x^n#
#=1+2sum_{n=1}^infty x^n# -
Since
#sinx=sum_{n=0}^infty(-1)^n{x^{2n+1}}/{(2n+1)!}# ,by replacing
#x# by#x^2# ,#Rightarrow f(x)=sum_{n=0}^infty(-1)^n{(x^2)^{2n+1}}/{(2n+1)!}# #=sum_{n=0}^infty(-1)^n{x^{4n+2}}/{(2n+1)!}#
Questions
Power Series
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Introduction to Power Series
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Differentiating and Integrating Power Series
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Constructing a Taylor Series
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Constructing a Maclaurin Series
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Lagrange Form of the Remainder Term in a Taylor Series
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Determining the Radius and Interval of Convergence for a Power Series
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Applications of Power Series
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Power Series Representations of Functions
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Power Series and Exact Values of Numerical Series
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Power Series and Estimation of Integrals
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Power Series and Limits
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Product of Power Series
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Binomial Series
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Power Series Solutions of Differential Equations