If we have a function $f(x)$, it is possible that we can express it in terms of an infinite linear combination of power functions of the form $ax^n$. But my question is that why we cannot express it in terms of finite linear combination of those power functions. Tag suggested.
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$\begingroup$ Do you want to give us an example of a function of which you believe should be expressable as a finite linear combination? $\endgroup$Moritz– Moritz2016-09-18 15:52:05 +00:00Commented Sep 18, 2016 at 15:52
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2 Answers
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Because take for instance
$$f(x)=e^x=\sum_{k=0}^\infty \frac {x^k}{k!}.$$
Then the function $f$ is expressed in terms of an infinite linear combination of power functions, but you can find a finite sum because it only works for polynomials, and $\exp$ is not a polynomial.
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No, it is not true that we can express every function as a power series. Try $f(x)=\begin{cases}1&x\in\Bbb Q\\0&x\notin\Bbb Q\end{cases}$.
answered Sep 18, 2016 at 15:50
Hagen von Eitzen
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