Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { x } { \sqrt { 9 + x ^ { 2 } } }$
A) $| x |

Question

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { x } { \sqrt { 9 + x ^ { 2 } } }$
A) $| x | < 1$
B) $| x | < 10$
C) $| x | < 6$
D) $| x | < 9$
E) $| x | < 3$

Quizplus · Accepted Answer

The function can be rewritten as $x (9 + x^2)^{-1/2}$. Using the binomial series expansion for $(1 + u)^n$ where $u = \frac{x^2}{9}$ and $n = -1/2$, the series converges when $|u| < 1$, which translates to $|x^2/9| < 1$ or $|x| < 3$.

Use the Binomial Series to Expand the Function as a Power