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Hi,
I've computed 512 terms of a power series numerically. Below are the first 20 terms.
$$
\begin{align*}
w(z)&=0.182456 0.00505418 z+0.323581 z^20.708205 z^30.861668 z^4+0.83326 z^5+0.994182 z^6 \\ &1.18398 z^70.849919 z^8+2.58123 z^90.487307 z^{10}7.57713 z^{11}+3.91376 z^{12}\\
&+19.6052 z^{13}12.717 z^{14}40.7957 z^{15}+40.3343 z^{16}+66.9766 z^{17}133.148 z^{18}\\
&78.9037 z^{19}+\cdots
\end{align*}
$$
And wish to show the radius of convergence is approx ##0.574342##. The Ratio test fails as successive ratios are finite but do not tend to a limit as shown in the plot below and I cannot use the Alternating Series test either or integral test or other convergence test I am aware of.
I was wondering if someone could suggest another method I could use to suggest experimentally that this series is appearing to converge (conditionally) in the indicated domain?
I've computed 512 terms of a power series numerically. Below are the first 20 terms.
$$
\begin{align*}
w(z)&=0.182456 0.00505418 z+0.323581 z^20.708205 z^30.861668 z^4+0.83326 z^5+0.994182 z^6 \\ &1.18398 z^70.849919 z^8+2.58123 z^90.487307 z^{10}7.57713 z^{11}+3.91376 z^{12}\\
&+19.6052 z^{13}12.717 z^{14}40.7957 z^{15}+40.3343 z^{16}+66.9766 z^{17}133.148 z^{18}\\
&78.9037 z^{19}+\cdots
\end{align*}
$$
And wish to show the radius of convergence is approx ##0.574342##. The Ratio test fails as successive ratios are finite but do not tend to a limit as shown in the plot below and I cannot use the Alternating Series test either or integral test or other convergence test I am aware of.
I was wondering if someone could suggest another method I could use to suggest experimentally that this series is appearing to converge (conditionally) in the indicated domain?
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