ratio test did I do it right?

[Jac8897]

1. The problem statement, all variables and given/known data
Hi
I want to see if you guys could check my work an tell me if i did it correct and also explain me how to cancel "look at the graph"


2. Relevant equations



3. The attempt at a solution

[NastyAccident]

Yes.

When in doubt, set a term an equal to what is being summed. In this case, an = 2^n/n!.

Now, get the next term in the series so an+1. In this case, an+1 = 2^(n+1)/(n+1)!.

Then, plug in an and an+1 into |\frac{a_{n+1}}{a_{n}}|

Now, the n! cancels because n! is equivalent to n*(n-1)(n-2)(n-3)(n-4). Whereas (n+1)!, is equivalent to (n+1)(n)(n-1)(n-2)(n-3)(n-4).

So, as you can see, the terms to the right of (n+1) are canceled out.

Sincerely,

NastyAccident

[Jac8897]

Yes.

When in doubt, set a term an equal to what is being summed. In this case, an = 2^n/n!.

Now, get the next term in the series so an+1. In this case, an+1 = 2^(n+1)/(n+1)!.

Then, plug in an and an+1 into |\frac{a_{n+1}}{a_{n}}|

Now, the n! cancels because n! is equivalent to n*(n-1)(n-2)(n-3)(n-4). Whereas (n+1)!, is equivalent to (n+1)(n)(n-1)(n-2)(n-3)(n-4).

So, as you can see, the terms to the right of (n+1) are canceled out.

Sincerely,

NastyAccident


thanks for the explanation was very helpful