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So I have been studying the convergence of Taylor series from Tom M Apostol. There is a theorem which states the sufficient condition for convergence of Taylor series. Quoting the theorem we've

Assume $f$ is infinitely differentiable in an open interval $I = (a - r, a + r)$, and assume that there is a positive constant $A$ such that $$ |f^{(n)}(x)| \le A^n$$ for $n = 1, 2, 3, \dots$ and every $x$ in $I$. Then the Taylor's series generated by $f$ at $a$ converges to $f(x)$ for each $x$ in $I$.

My question is how to arrive at $|f^{(n)}(x)| \le A^n$ deductively ? I tried looking it up on the internet but haven't found anything that solves my issue.

Thanks!

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  • $\begingroup$ Do you mean how to deduce that such a bound would be a sufficient condition for convergence? It is a strange question asking how to deduce a hypothesis. Hypotheses are starting points. One could guess, or motivate a hypothesis. In the case of necessary and sufficient conditions one could deduce it, but this one is only a sufficient condition. $\endgroup$ – logarithm May 13 at 13:13
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The assertion $$ |f^n(x)| \le A^n $$ is a hypothesis in this theorem. You can't "arrive at it deductively" in general. When it happens to be true for any particular function $f$ then you can conclude something about the Taylor series for that function.

For example, the $n$th derivatives of $\sin$ are $\sin$ and $\cos$. Their absolute values are always at most $1$. This theorem then implies that the Taylor series for $\sin$ converges everywhere to the value of the function.

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