Suppose the power series $\sum_{0}^{\infty}c_nx^n $ converges in $(-r,r)$, I want to know if:
$|\sum_{0}^{\infty}c_nx^n| \leq c|x|^N$ for all $x \in(-r,r)$, where c is a constant, then $a_0=a_1=...=a_{N-1}=0$ ?
power series $|\sum_{0}^{\infty}c_nx^n| \leq c|x|^N$ implies $a_0=a_1=...=a_{N-1}=0$
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real-analysis
power-series
1 Answers
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Dividing by $|x|^N$ in the inequality you have $$\left|\sum_{n\geq 0} a_n x^{n- N}\right| \leq c $$ note that $n-N \leq 0$ for $n\leq N$ and if one of $a_0 , ... ,a_{N-1}$ is non-zero, then you can choose a $x\in (-r,r)$ small (need to find it) such that $$\left|\sum_{n\geq 0} a_n x^{n- N}\right|>c$$ because the terms $x^{n-N}$ goes to "infinity" when $x \to 0$ for $n-N<0$, this is a contradiction, thus $a_0 = ... =a_{N-1}= 0$.