# Thread: Need help on series! Relatively easy...

1. ## Need help on series! Relatively easy...

Alright. We were just assigned this work and its due tomorrow. I got everything except these three problems, so help would be appreciated.

1) Find the radius of convergence and interval of convergence of the series:

((-1)^n)((X^2n)/(2n)!)

does this turn into lim as n->infinity of 2n/(2n+1) times |x^2| ?

The ! threw me off.

2) If R = infinity, does the series converge at only the x? For example...
lim n-> infinity of 24n/6 |X-4| would the interval be only 4 ?

3) This is the main problem ive been struggling on...
Suppose that the series of CnX^n converges when x=-4 and diverges when x=6. What can be said about the convergence or divergence of the following series?

a) Series (to infinity from n=0) of Cn,
b) Series of Cn8^n
c) Series of Cn(-3)^n
d) Series of ((-1)^n)Cn9^n

Any and all help is appreciated.

2. Alright. We were just assigned this work and its due tomorrow. I got everything except these three problems, so help would be appreciated.

1) Find the radius of convergence and interval of convergence of the series:

((-1)^n)((X^2n)/(2n)!)

does this turn into lim as n->infinity of 2n/(2n+1) times |x^2| ?

The ! threw me off.
This looks like the Taylor series for cos(x).

$\displaystyle cos(x)=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+.......=\sum_{k=0}^{\infty}(-1)^{k}\frac{x^{2k}}{(2k)!}$

Compare to yours.

3. Originally Posted by 3deltat
Alright. We were just assigned this work and its due tomorrow. I got everything except these three problems, so help would be appreciated.

1) Find the radius of convergence and interval of convergence of the series:

((-1)^n)((X^2n)/(2n)!)

does this turn into lim as n->infinity of 2n/(2n+1) times |x^2| ?

The ! threw me off.
Applying the ratio test gives

\displaystyle \begin{aligned}\lim_{n\to\infty}\left|\frac{(-1)^{n+1}x^{2n+2}}{(2n+2)(2n+1)(2n)!}\cdot\frac{(2n )!}{(-1)^nx^{2n}}\right|&=|x^2|\cdot\lim_{n\to\infty}\le ft|\frac{1}{(2n+1)(2n+2)}\right|\\ &=|x|\cdot{0}<1~~\forall{x}\in\mathbb{R}\end{align ed}

2) If R = infinity, does the series converge at only the x? For example...
lim n-> infinity of 24n/6 |X-4| would the interval be only 4 ?
Suppose you have a power series $\displaystyle \sum{a_n(x-c)^n}$ and it converges only at its center c then the Interval of Convergence is $\displaystyle x\in\left\{c\right\}$ and its radius of convergence is $\displaystyle R=0$
3) This is the main problem ive been struggling on...
Suppose that the series of CnX^n converges when x=-4 and diverges when x=6. What can be said about the convergence or divergence of the following series?

a) Series (to infinity from n=0) of Cn,
b) Series of Cn8^n
c) Series of Cn(-3)^n
d) Series of ((-1)^n)Cn9^n
I suppose you mean this, $\displaystyle \sum{c_nx^n}$ converges on $\displaystyle x\in[-4,6)$

Then all your questions can be answered by evaluating of the series in question's value for x is within the interval of convergence.