Question

write the following functions as power series, and calculate their open intervals of convergence
\(\frac{\sin(x^{3})}{x}\)

Answer 1

\[\sin{x}=\sum_{n=0}^\infty (-1)^{n}\frac{x^{2n+1}}{(2n+1)!}\\
\sin{x^3}=\sum_{n=0}^\infty (-1)^{n}\frac{x^{6n+3}}{(2n+1)!}\\
\frac{\sin{x^3}}{x}=\sum_{n=0}^\infty (-1)^{n}\frac{x^{6n+2}}{(2n+1)!}\]
Since the R of sin(x) is infinity, then the R of this is also infinity.

Answer 2

thank you very much