Find the interval of convergence question.

Question

Find the interval of convergence for the power series

∑n=0∞ ( (−2)^n / n! ) * (x−3)n

Give your answer using interval notation. If you need to use INF. If there is only one point in the interval of convergence, the interval notation is [a]. For example, if 0 is the only point in the interval of convergence, you would answer with [0].

Answer 1

∑₀∞(−2)ⁿ / n! ) * (x−3)ⁿ.

lim n→∞ I(-2)ⁿ⁺¹(x - 3)ⁿ⁺¹·n!/[(-2)ⁿ(n + 1)!(x-3)ⁿ]I = lim n→∞ I2(x-3I/(n+1)I = 0 < 1.

So, series converges for any x: - ∞ < x < ∞