Derivative and integral of a power series proof (not getting a step)

[Ryker]

Homework Statement

Basically, there's no problem statement per se, I'm just trying to understand the proof that the following sequences have the same radius of convergence:

$$(1) \displaystyle\sum_{k=0}^{\infty} c_{k}x^{k}$$
$$(2) \displaystyle\sum_{k=0}^{\infty} kc_{k}x^{k-1}$$
$$(3) \displaystyle\sum_{k=0}^{\infty} c_{k}\frac{x^{k+1}}{k+1}$$

The Attempt at a Solution

The proof begins by saying that if (2) converges absolutely, then since $$|c_{k}x^{k}| = |x||c_{k}x^{k-1}| \leq |kc_{k}x^{k-1}|$$ the comparison test implies the same for (1).

I don't get the inequality part. I mean, since k is a natural number, hence part of an unbounded set, this must be true for all k greater than some k₀, but am I missing something that would render this true for all k? I mean, as the proof is stated, nothing suggests that it's supposed to be interpreted as I interpreted it, but if it holds true for all k, then I don't see how this is so.

Any help with understanding this would, as always, be greatly appreciated.

 

[I like Serena]

You're exactly right.
It won't hold for any k, but just for k that are large enough.

However, that is sufficient to determine whether the series converges or not.

This should be mentioned in the proof, so if it's not, that means that they've been sloppy.

 

[Ryker]

You're exactly right.
It won't hold for any k, but just for k that are large enough.

Thanks, great to know I was on the right track.

However, that is sufficient to determine whether the series converges or not.

Yeah, I know, that's why I first just wanted to let it slide and move on, but after a while decided to check with the PF community anyway, just to see I'm not missing something

This should be mentioned in the proof, so if it's not, that means that they've been sloppy.

Usually the professor that made these lecture notes would mention something like that, which is what made it all the more puzzling when in this case he didn't. So my first thought was that I'm just not seeing where the step stems from.

I sometimes feel silly making these threads for such small things, but I'm just doing some Maths on my own during summer, and I hate it when I don't get all the steps in a proof or something. I feel I can't just move on, and there's also no one else I could really ask at this time.

 

[I like Serena]

Glad to have been of help!