Final answer:
The power series converges and has a radius of convergence of 10, determined by using the ratio test.
Step-by-step explanation:
The power series ∑ [infinity]ₙₜ₁ (0.1)^ₙx^ₙ is examined for convergence using the ratio test. We consider the absolute value of the ratio of the n+1 term to the n term:
|a_(n+1) / a_n| = |(0.1)^(n+1)x^(n+1) / (0.1)^ₙx^ₙ| = 0.1|x|
The series converges if this ratio is less than 1, hence it converges for |0.1x| < 1. This simplifies to |x| < 10, which means that the radius of convergence is 10.