Verified answer

Use the ratio test.

r = lim(n→∞) |[10^(n+1) (x - 2)^(n+1) / (n+1)!] / [10^n (x - 2)^n / n!]|

..= 10|x - 2| * lim(n→∞) 1/(n+1)

..= 0.

Since r = 0 < 1 for all x, this series converges for all real x.

I hope this helps!