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# Find the series’ radius and interval of convergence. For what values of x does the series converge?$\sum _ { n = 0 } ^ { \infty } x ^ { n }$

Solution

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At the beginning of the task, in order to determine the radius of convergence of the given order, we will use The Ratio Test which says the following:

Let $\sum u_n$ be an arbitrary series such that exists:

\begin{align}\lim_{n\to\infty}\frac{|u_{n+1}|}{|u_n|}=\rho\end{align}

Then the following applies:

• (i) If $\rho<1$, then the series converges absolutely.
• (ii) If $\rho>1$ (or $\rho$ is infinite), then the series diverges.
• (iii) If $\rho=1$, then the test is inconclusive.

Note: If the powers series is given as

$\sum_{n=0}^{\infty} a_n(x-c)^n,$

then

$u_n=a_n(x-c)^n.$

How will we determine $u_n$?

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