Differentiating and Integrating Power Series
Key Questions
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If
#f(x)=sum_{k=0}^infty c_kx^k# , then
#f^{(n)}(x)=sum_{k=n}^infty k(k-1)(k-2)cdots(k-n+1)c_kx^{k-n}# By taking the derivative term by term,
#f'(x)=sum_{k=1}^infty kc_kx^{k-1}#
#f''(x)=sum_{k=2}^infty k(k-1)c_kx^{k-2}#
#f'''(x)=sum_{k=3}^infty k(k-1)(k-2)c_kx^{k-3}#
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#f^{(n)}(x)=sum_{k=n}^infty k(k-1)(k-2)cdots(k-n+1)c_kx^{k-n}# -
If
#sum_{n=0}^infty c_n x^n# is a power series, then its general antiderivative is#intsum_{n=0}^infty c_n x^n dx=sum_{n=0}^infty c_n/{n+1}x^{n+1}+C# .(Note that integration can be done term by term.)
I hope that this was helpful.
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One of the most useful properties of power series is that we can take the derivative term by term. If the power series is
#f(x)=sum_{n=0}^inftyc_nx^n# ,then by applying Power Rule to each term,
#f'(x)=sum_{n=0}^infty c_n nx^{n-1}=sum_{n=1}^inftync_nx^{n-1}# .(Note: When
#n=0# , the term is zero.)I hope that this was helpful.
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