Largest open interval of convergence for the power series about c=0 , represents the function f(x)=2/(x^2 +4)?

What is the largest open interval of convergence for the power series about c=0 which represents the function f(x)=2/(x^2 +4)?





(don't worry about endpoints)





possible solutions...





(-1/6, 1/6)





(-1/4, 1/4)





(-1/3, 1/3)





(-1/2, 1/2)





(-1, 1)





(-3/2, 3/2)





(-2, 2)





(-3, 3)





(-4, 4)





or is it none of these

Largest open interval of convergence for the power series about c=0 , represents the function f(x)=2/(x^2 +4)?
(-2,2).





You factor out the 4 to get





2/(x^2 +4)= 2/4*1/(x^2/4+1) = 2/4* 1/[(x/2)^2+1] so you need abs(x/2)%26lt;1 from the arctan formula, so abs(x)%26lt;2. The endpoints you check separately (or recall from the arctan formula).

strawberry