Prove Weierstrass M-test.Prove by induction that the function f(x)=exp(-1x2) for x0 and f(0)=0 has derivatives of all orders at every point in R and that all of these derivatives vanish at x=0. Hence this function is not given by its Taylor expansion about x=0.Prove that if n=0anxn converges for x=x0 and diverges for x=x1, thena.n=0anxn converges absolutely for |x|<|x0|, andb.n=0anxn diverges for |x|>|x1|.How could one use the Ratio Test to establish criteria for the radius of convergence?Suppose that ??an diverges and that {an} is bounded. Prove the radius of convergence of the power series ??anxn is equal to 1.Write the Taylor series for 1-x2 centered at 0. Prove that the series converges to 1-x2 for xin(-1,0].