Find the radius of convergence of the power series consult

Question

1. Consult a list of common power series to find the exact sum of the following series:

a. 1 - 1/3 + 1/5 - 1/7 + ··· + (-1)ⁿ (1/2n+1) + ···

b. _n=0Σ^∞ ((-1)ⁿπ²ⁿ/3²ⁿ(2n!))

2. Apply the Ratio Test to determine whether _n=1Σ^∞ (3ⁿ/n2²ⁿ⁺¹) converges.

3. Apply the Root Test to determine whether _n=1Σ^∞ (3n-1/2n+5)ⁿ converges.

4. Find the radius of convergence of the power series.

a. _n=0Σ^∞ (n!xⁿ/2ⁿ)

b. _n=0Σ^∞ (x²ⁿ/(2n)!)