Power series - Different problem

[Neon32]

In the power series below, I've used the ratio test and at the end I got |x-2| times infinity which is >1 so it diverges.. and in this case there is no interval of convergence because it's times inifnity.. How did he conclude that it converges at x=2??

 

[Mark44]

In the power series below, I've used the ratio test and at the end I got |x-2| times infinity which is >1 so it diverges.. and in this case there is no interval of convergence because it's times inifnity.. How did he conclude that it converges at x=2??

View attachment 115211

Because at x = 2, the limit is mulitplied by 0, and the part that says "when x ≠ 0" really should say "when x ≠ 2".

I should add that the absolute value on the fraction inside the limit is unnecessary. All the terms are positive, so the | | signs can be removed.

And, the writer's grasp of English is not very good. It should say, "The series is convergent at ..." or "The series converges at ..."

 

[Neon32]

Because at x = 2, the limit is mulitplied by 0, and the part that says "when x ≠ 0" really should say "when x ≠ 2".

I should add that the absolute value on the fraction inside the limit is unnecessary. All the terms are positive, so the | | signs can be removed.

And, the writer's grasp of English is not very good. It should say, "The series is convergent at ..." or "The series converges at ..."

I'm not sure if I got the first line well. if x=2 the limit will be multiplied by 0 so at the end we'll get 0 times inifnity. How is that convergent? Please elaborate. Thanks for your help!

 

[Mark44]

Here's the series:
$$\sum_{n = 1}^\infty \frac{(2n + 1)!}{n^3}(x - 2)^n$$
If x = 2, every term in the series is 0, so the sum of the series is 0, and it is therefore convergent.