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I am likely just missing something fundamental here, but I recently just revisited series and am looking over some notes.

In my notes, I have written that if

## \lim_{x \to +\infty} \frac{a_{n+1}}{a_n} = L ##

Then ## | x - x_o | = 1/L ##

But shouldn't the correct expression be $$ | x - x_o | = L $$ ?

Why is the radius of convergence 1/L instead of L? I was under the assumption since

## \frac{(x - x_o)^{n+1}} {(x-x_o)^n} = x - x_o ##

that this term (absolute value) would be less than and/or (possibly) equal to L? Are my notes inaccurate or is there a reason for R being 1/L?

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# Radius of convergence derivation

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