On what interval are you guaranteed that the expansion of Log[x] in powers of (x

Question

On what interval are you guaranteed that the expansion of
Log[x] in powers of (x - 2) converges to Log[x]?

The answer should be (0,4)... I think.
My work is...
Convergence intervals:
b - R to b + R

R is the shortest distance between expansion center and nearest singularity.
Here, b > 0.
We know Log[x] fails at x = 0.

What is R?
For this situation,
R = 0
is the magic number.
When you expand f[x] in powers of (x- 2), then:
-> The approximations in powers of (x- 2) will hold up beautifully inside the interval
(2 - R, 2 + R) = (2 -0,2+0),
-> But the approximations in powers of (x - 1) will break down outside
[2 - R, 2 + R] = [2 -0,2+0].
This is why you ran into barriers at
x=2 and x=2.

What did I do wrong? And why?

Explanation / Answer

Convergence intervals: b - R to b + R R is the shortest distance between expansion center and nearest singularity. Here, b > 0. We know Log[x] fails at x = 0. What is R? For this situation, R = 0 is the magic number. When you expand f[x] in powers of (x- 2), then: -> The approximations in powers of (x- 2) will hold up beautifully inside the interval (2 - R, 2 + R) = (2 -0,2+0), -> But the approximations in powers of (x - 1) will break down outside [2 - R, 2 + R] = [2 -0,2+0]. This is why you ran into barriers at x=2 and x=2. Your work is absolutely fine..

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