Derivatives and Integrals of Power Series
As long as we are strictly inside the interval of convergence, we can take derivatives and integrals of power series term by term. Namely,
ddx(∞∑n=0anxn)=∞∑n=1nanxn−1
∫(∞∑n=0anxn)dx=(∞∑n=0anxn+1n+1)+C
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The same holds for power series centered at a.
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