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(Solved): Prove Weierstrass M-test. Prove by induction that the function f(x)=exp(-(1)/(x^(2))) for x!=0 and f ...
Prove Weierstrass M-test. Prove by induction that the function
f(x)=exp(-(1)/(x^(2)))for
x!=0and
f(0)=0has derivatives of all orders at every point in
Rand 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
\sum_(n=0)^(\infty ) a_(n)x^(n)converges for
x=x_(0)and diverges for
x=x_(1), then a.
\sum_(n=0)^(\infty ) a_(n)x^(n)converges absolutely for
|x|<|x_(0)|, and b.
\sum_(n=0)^(\infty ) a_(n)x^(n)diverges for
|x|>|x_(1)|. How could one use the Ratio Test to establish criteria for the radius of convergence? Suppose that
\sum a_(n)diverges and that
{a_(n)}is bounded. Prove the radius of convergence of the power series
\sum a_(n)x^(n)is equal to 1 . Write the Taylor series for
\sqrt(1-x)centered at 0 . Prove that the series converges to
\sqrt(1-x)for
xin(-1,0].
