series' radius of convergence

[idle sigil]

hi. can anyone explain how to calculate the radius of convergence for any series? im too bored to research it on my own lol

[onyx forum]

context: real or complex analysis?

if you have a formula for all the terms, you just want to check what distances passes the ratio test for convergence

the bonus in complex analysis is that this radius is related to how far the singularities are

[broken flower]

or equivalently use Hadamard formula

[idle sigil]

how might one reach the maximum range for which the series converges? also, im mostly looking for real analysis. ive not studied enough complex analysis for this question to come up

[idle sigil]

ive never heard of this

[broken flower]

it's given by
$$R=\frac{1}{\limsup_{k\to\infty} |a_k|^{1/k}}$$

[raven estuary]

artemetra

[broken flower]

but it's equivalent to the ratio test Element118 was describing

i just like this one because it's an explicit formula in which you can just plug stuff in

[broken flower]

it would be helpful if you provided an example so we can show you

[hollow tinsel]

the maximum range / the interval of convergence (IOC) is found after you have your radius, thus the endpoints, then by testing of those individual endpoints converge or not

.. say you have series from n to inf of (x-1)^n, plug endpoint in for x to determine if it converges at that point, then change the inequality sign to either include it or not

[idle sigil]

i honestly dont have any examples. I was just curious because people always just use the radius of convergence without proving it, and i never got to learn the proof. from what youve shown, i'll need more knowledge in calculus to truly understand how its calculation works. But thank you for explaining

.solved