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For example, I define $\sin x=\sum_{k=0}^\infty\frac{(-1)^kx^{2k+1}}{(2k+1)!}$, should I show that $\sum_{k=0}^\infty\frac{(-1)^kx^{2k+1}}{(2k+1)!}$ is convergent for all real $x$ first, or I can define it first even I haven't proved it's convergent (e.g. in writing a thesis)? Thanks.

Thomas Andrews
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JSCB
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    While you are using the fact that the series converges to make sense of the definition, whether or not you should prove it (or present a proof) is highly dependent on context we don't have. Often in mathematical writing, "well-known" facts can be used without explicit justification, or by brief citation. Do you have a supervisor whom you could ask about expected standards? – Jonas Meyer Aug 25 '13 at 03:07

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Yes, for the $x$ for which you assert it is defined, you must show the series converges. For your example, you'd need to prove that it converges for all $x$.

You could potentially say something like:

$$f(x)=\sum_{i=0}^\infty a_i x^i$$ for all values $x$ for which it converges. But then you only know that $f(0)=0$ - you have no idea what the domain of your function is.

A "definition" of a function ascribes a value output for each possible input. If you don't know the series converges for some value $x$, then you haven't defined it for that value.

Thomas Andrews
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    Ah, so I can first have a theorem like "$\sum \cdots$ is convergent" and give a proof, then I define $\sin x=\sum\cdots$ ? – JSCB Aug 25 '13 at 02:50
  • Yes, that's the essence. You can't define a function without knowing the domain of the function. – Thomas Andrews Aug 25 '13 at 02:52