Finding Series Radius and Interval of Convergence

[k1ll3rdr4g0n]

I am hopelessly confused on a homework assignment.

The problem says " (a) Find the series radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?"

Attached is a sample of problems from the book.

Any help would be appreciated!

 

[sutupidmath]

These are power series. The radius of convergence is

$$\lim_{x\to\infty}\left|\frac{a_n_+_1}{a_n}\right|$$ then the series converges for


$$\lim_{x\to\infty}\left|\frac{a_n_+_1}{a_n}\right|<1$$

After that you find that the series converges say for x in the interval (a,b) and after that try to test whether the series converges at a and b, by letting x=a, and x=b respectively.

 

[HallsofIvy]

The series converges absolutely inside the radius of convergence, diverges outside and may converge absolutely, converge conditionally, or diverge at the endpoints. That's why you have to test those separately.