If the power series $\sum C_{n} x^{n}$ converges for $x=a, a0$ , then it converges for $x=\frac{a}{2}$ .

Name: If the power series $\sum C_{n} x^{n}$ converges for $x=a, a>0$ , then it converges for $x=\frac{a}{2}$ .
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Question

If the power series $\sum C_{n} x^{n}$ converges for $x=a, a>0$ , then it converges for $x=\frac{a}{2}$ .

Quizplus · Accepted Answer

If a power series converges for some $x=a$, where $a>0$, then by the properties of power series, it also converges absolutely for any $x$ such that $|x| < |a|$. Since $\left|\frac{a}{2}\right| < |a|$, the series converges for $x=\frac{a}{2}$.

If the Power Series ∑Cnxn\sum C_{n} x^{n}∑Cn​xn Converges For x=a,a>0x=a, a>0x=a,a>0 , Then It Converges For

If the Power Series $\sum C_{n} x^{n}$ Converges For $x=a, a>0$ , Then It Converges For