Suppose $p\leqslant a_n\leqslant q$ $\forall$ $n\geqslant 1$, where $p, q \in \mathbb{R}$. Then how to calculate the radius of convergence of $$\sum_{n=0}^\infty a_nx^n.$$ I tried using ratio test, root test. Any idea will be very helpful.
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comparison test? – Angina Seng Jul 24 '17 at 06:00
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I do not know how to get radius of convergence using comparison test. – TRUSKI Jul 24 '17 at 06:01
3 Answers
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If $0<p\leqslant q$, then the radius of convergence is $1$, because $\sqrt[n]p\leqslant\sqrt[n]{a_n}\leqslant\sqrt[n]q$ and $\lim_{n\in\mathbb N}\sqrt[n]p=\lim_{n\in\mathbb N}\sqrt[n]q=1$.
José Carlos Santos
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Hint:$$\sum_{n=0}^{\infty}a_nx^n \leq \sum_{n=0}^{\infty}\max\{|p|,|q|\}x^n=\max\{|p|,|q|\}\sum_{n=0}^{\infty}x^n$$
dromastyx
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You can only achieve a lower bound, since $a_n$ may be $0$ for all $n$. $$| \sum a_n x^n | \le \max\{|q|,|p|\} \sum x^n $$ which will certainly converge for $|x| < 1$. Hence your radius of convergence is greater or equal to $1$.
crankk
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