Power series

[Guest]

Please help me
Thank you

[Guest]

Where did you get this problem? Are you taking a course? If so you should know several "convergece tests" you could use! My first thought was to use the "ratio test". That only applies to series of postive numbers so we would first take the absolute value to check for "absolute convergence". Of courses, if this series converges "absolutely" then it converges!

[tex]\frac{\frac{x^{2k+3}}{2k+3}}{\frac{x^{2k+1}}{2k+1}}= \frac{2k+ 1}{2k+3}\frac{x^{2k+ 3}}{x^{2k+1}}= \frac{2k+ 1}{2k+ 3}x^2[/tex].

The limit, as k goes to infinity is [tex]x^2[/tex] which is less than 1 only for -1< x 1. And it is larger than 1 for x> 1 or x< -1. So, no, this series does NOT converge for all -9< x< 9.