I have a doubt concerning concerning power series comparison.
There are two power series $f$ and $g$ with the coefficients $(a_i)_{i \in \mathbb{N}}$ and $(b_i)_{i \in \mathbb{N}}$. That is saying :
$$f(t) = \sum_{i=0}^\infty a_i t^i$$
and same thing for $g$ with the $b_i$. We know that both $f$ and $g$ are greater than $0$ on $\mathbb{R}_+$.
Can we say that if $ \forall i, |a_i| > |b_i|$,so we have $\forall t > 0, f(t) > g(t)$ ?
Could anyone answer this question by a reference or a proof ?
If it is not true, what conditions let this sentence be checked ?
Thank you in advance !
No that's not enough. Consider the power series $\sum_{i = 0}^\infty (-1)^i\frac{2^i}{i!} t^i$ and $\sum_{i = 0}^\infty \frac{1}{i!} t^i$ of $e^{-2t}$ and $e^t$. Obviously $|(-1)^i\frac{2^i}{i!}| = \frac{2^i}{i!} > \frac{1}{i!} = |\frac{1}{i!}|$ but $e^{-2t} < e^t$ for all $t > 0$.