What Happens When Your Comparison Test Fails or Is Negative?

[Sabricd]

what happens if your comparison test fails and you limit comparison test is negative or zero? Does that ever happen?

 

[tiny-tim]

Hi Sabricd!

If you choose the wrong comparison, that's bound to happen.

Is there a particular series that's worrying you? ​

 

[Sabricd]

Hello!

Yes! This is probably silly but I am not sure how to simplify...

=1 + 2/(1!)*x + 6/(2!)*x^2 + 24/(3!)*x^3 + 120/(4!)*x^4 ...

I am trying to solve a Maclaurin series but I don't have that much experience with factorials. I would really REALLY appreciate if you could give me some insight!

Thanks!

 

[micromass]

Your series is actually

$$1+2x+3x^2+4x^3+...+(n+1)x^n+...$$

thats a little simpler, isn't it?

 

[Sabricd]

Yes!
Sorry I'm still kind of lost. I'm teaching myself Cal2..but I cannot understand how that is derived. Would you mind explaining to me how you simplified it?

Thanks!

 

[micromass]

Well, the numerators of your series are 1,2,6,24,120,... The general term is certainly (n+1)!
So your series becomes $$\frac{(n+1)!}{n!}x^n$$. Working this out gives you the answer.

 

[Sabricd]

Ok thanks! :)