<s>[INST] Examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$ How to examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$ for $a, b, c> 0$ using Taylor's theorem?
 [/INST] You have, from Taylor expansion of $e^u$ around $0$ to second order,
$$\begin{align}
\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2} &= 
e^{\frac{1}{n}\ln a} - \frac{1}{2}\left(e^{\frac{1}{n}\ln c}+e^{\frac{1}{n}\ln b}\right)\\ &=
1+\frac{1}{n}\ln a + \frac{1}{2n^2}\ln ^2 a \\&\qquad- \frac{1}{2}\left(2+\frac{1}{n}\ln b + \frac{1}{2n^2}\ln ^2 b+\frac{1}{n}\ln c + \frac{1}{2n^2}\ln ^2 c\right) + o\left(\frac{1}{n^2}\right) \\
&= \frac{1}{2n}(2\ln a - (\ln b + \ln c)) + \frac{1}{4n^2}(2\ln ^2 a - ( \ln ^2 b+\ln ^2 c)) + o\left(\frac{1}{n^2}\right).
\end{align}$$
If $(2\ln a - (\ln b + \ln c)) \neq 0$, can you conclude (by theorems of comparisons)? And if $(2\ln a - (\ln b + \ln c)) = 0$?
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