# Is the (x-a) format necessary when finding the radius of convergence for a geometric series?

So I am taking AP Calculus BC, and we are currently working on convergence and divergence of series. I came across the following problem in one of my homework assignments:

Here is the work I did to find the radius of convergence:

My teacher said I shouldn't have added 5 to each side, and left it at the highlighted step, meaning 6 would be the radius of convergence. I asked him why, and he said that the -5 is what centers the function.

Why is this the case? Does it have to do with Taylor Polynomials at all? If I do add 5, then isn't the radius still 6 because (11--1)/2 = 6

If any of you have an intuitive explanation for this, I would really appreciate it!

• Personally I don't think the format matters. I think your teacher asked you to leave it at that step just because that format is the most convenient way to recognize the radius. This might just be a preference about writing solutions. – xbh Dec 9 '18 at 15:36
• You are perfectly right, You may very well add $5$ and the radius is still $6$ as You properly explained ($\frac{11-(-1)}{2}=6$) – Peter Melech Dec 9 '18 at 15:36
• @PeterMelech Thanks! Do you think this would work for non-geometric series? – Addison Dec 9 '18 at 15:39
• For the general case the radius of convergence is given by "Cauchy-Hadamard" :$r=\frac{1}{\lim\sup_{n\rightarrow\infty}\sqrt[n]{|a_n|}}$ – Peter Melech Dec 9 '18 at 15:44
• @Addison Sorry for the sloppy comment above, but if we are talking about power series, You can always trace it back to a geometric series and this is where this formula by Cauchy-Hadamard comes from – Peter Melech Dec 9 '18 at 15:57