Differentiating and Integrating Power Series
Key Questions

If
#f(x)=sum_{k=0}^infty c_kx^k# , then
#f^{(n)}(x)=sum_{k=n}^infty k(k1)(k2)cdots(kn+1)c_kx^{kn}# By taking the derivative term by term,
#f'(x)=sum_{k=1}^infty kc_kx^{k1}#
#f''(x)=sum_{k=2}^infty k(k1)c_kx^{k2}#
#f'''(x)=sum_{k=3}^infty k(k1)(k2)c_kx^{k3}#
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#f^{(n)}(x)=sum_{k=n}^infty k(k1)(k2)cdots(kn+1)c_kx^{kn}# 
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.

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^{n1}=sum_{n=1}^inftync_nx^{n1}# .(Note: When
#n=0# , the term is zero.)I hope that this was helpful.
Questions
Power Series

Introduction to Power Series

Differentiating and Integrating Power Series

Constructing a Taylor Series

Constructing a Maclaurin Series

Lagrange Form of the Remainder Term in a Taylor Series

Determining the Radius and Interval of Convergence for a Power Series

Applications of Power Series

Power Series Representations of Functions

Power Series and Exact Values of Numerical Series

Power Series and Estimation of Integrals

Power Series and Limits

Product of Power Series

Binomial Series

Power Series Solutions of Differential Equations