Question

AP Calculus help #5 convergence intervals

Answer 1

It is a geometric series with $r= \dfrac{2}{x^2+1}$ and it converges if and only if $|r|\ \textless \ 1$ so:

$|r|\ \textless \ 1\\\\\\\left|\dfrac{2}{x^2+1}\right|\ \textless \ 1\\\\\\\dfrac{2}{x^2+1}\ \textless \ 1\\\\\\2\ \textless \ x^2+1\\\\1\ \textless \ x^2\qquad|\sqrt{(\ldots)}\\\\\boxed{x\ \textless \ -1\qquad\vee \qquad x\ \textgreater \ 1}$

Answer d).