Power Series

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Introduction to Analysis › Power Series

Questions 1 - 1

Determine whether the following statement is true or false:

Some power series such as have a possible radius of convergence that can be found by computing the roots of the coefficients of .

False

True

This statement is false because every power series has a radius of convergence that is found by computing the roots the series coefficients.

The proof of one case is as follows:

When $x\ \epsilon\ \mathbb{R}$ is fixed, $x\neq x_0$ , and $\rho:=\frac{1}{\lim \text{sup}_{k\rightarrow \infty}|a_k|^{\frac{1}{k}}}$

with the assumption that $\frac{1}{\infty}=0$ and $\frac{1}{0}=\infty$ the Root Test is applied to the series .

When $\rho =0$ by the assumption and the notation that

$r(x):=\lim \sup_{k\rightarrow \infty}|a_k(x-x_0)^k|^{\frac{1}{k}}=|x-x_0\cdot \lim \sup_{k\rightarrow \infty}|a_k|^\frac{1}{k}$

implies that $r(x)=\infty>1$ therefore by the Root Test, does not converge for any $x\neq x_0$ and thus the radius of convergence of is $R=0=\rho$ .