I need some peer review of this problem

[RadiationX]

find the radius and interval of convergence of:

$$\sum_{n=1}^\infty\frac{(-1)^{(n+1)}nx^n}{2^n}$$

the radius is $$x<2$$

and the interval is [-2,2)

are these two answers correct?

 

[TenaliRaman]

Yes

-- AI
[edit]
wait a minute
just noticed something
u sure it converges for -2?
the interval should (-2,2)
[/edit]

 

[RadiationX]

when i put this in my calculator it adds up at [-2,2).

 

[shmoe]

when i put this in my calculator it adds up at [-2,2).

Step 1) throw away your calculator

Step 2) check convergence at x= -2 with pencil and paper

 

[RadiationX]

Step 1) throw away your calculator

Step 2) check convergence at x= -2 with pencil and paper

ah i used the alternating series convergence test incorrectly. by the nth term test for divergence i get that the limit of the series is $$\infty$$ which is not equal to zero so it diverges.

 

[RadiationX]

have i come to the correct conclusion shmoe?

 

[HallsofIvy]

At x= 2, the general term is $$(-1)^{n+1}n$$
At x= -2, the general term is $$(-1)^{2n+1}n$$

The series does not converge at either of those because the term does not go to zero. The interval of convergence is (-2, 2).