Use ratio test to find radius and interval of convergence of power series

[marylou]

Homework Statement

Use the ratio test to find the radius of convergence and the interval of convergence of the power series:

[[Shown in attachment]]

Homework Equations

a_n+1/a_n=k

Radius of convergence = 1/k

Interval of convergence: | x-a |∠ R

The Attempt at a Solution

I began by finding the summation which I concluded was:

Ʃ (2^n)(x-0)^n / n(factorial)

So a_n+1/a_n = [2ⁿ⁺¹/(n+1)(factorial)] *times* [ n(factorial)/ 2ⁿ ]

After cancelling, I arrived at 2/(n+1)

If that lim n→∞ is taken, it would be 0, meaning the radius of convergence is ∞, and the interval is from -∞ to ∞. Is that true or did I make a mistake, because the website for my homework won't allow me to enter infinity (∞)?

 

[Mark44]

Homework Statement

Use the ratio test to find the radius of convergence and the interval of convergence of the power series:

[[Shown in attachment]]

Homework Equations

a_n+1/a_n=k

Radius of convergence = 1/k

Interval of convergence: | x-a |∠ R

The Attempt at a Solution

I began by finding the summation which I concluded was:

Ʃ (2^n)(x-0)^n / n(factorial)

So a_n+1/a_n = [2ⁿ⁺¹/(n+1)(factorial)] *times* [ n(factorial)/ 2ⁿ ]

After cancelling, I arrived at 2/(n+1)

What happened to x?

BTW, the symbol for factorial is !, so you can write n! instead of n factorial.

If that lim n→∞ is taken, it would be 0, meaning the radius of convergence is ∞, and the interval is from -∞ to ∞. Is that true or did I make a mistake, because the website for my homework won't allow me to enter infinity (∞)?