Question

Need help understanding why my answer is wrong on a direct comparison test.

Answer 1

\[\sum_{0}^{\infty}\frac{ 1 }{ n! }\]

Answer 2

I wonder what you could compare this to...

Answer 3

I saw the n factorial terms as\[\frac{ 1 }{ n }+\frac{ 1 }{ n-1 }+\frac{ 1 }{ n-2 }+.....\frac{ 1 }{ 2 }+\frac{ 1 }{ 1 }\] then compared that to \[\frac{ 1 }{ n }\]which is a divergent p-series. Is it wrong to say that \[\frac{ 1 }{ n } \le \frac{ 1 }{ n! }\]????

Answer 4

Whoa there, where did you get \[\frac{ 1 }{ n }+\frac{ 1 }{ n-1 }+\frac{ 1 }{ n-2 }+.....\frac{ 1 }{ 2 }+\frac{ 1 }{ 1 }\]?

Answer 5

I goofed I am thinking multiplication

Answer 6

So... you found your error? :D

Answer 7

so maybe i Have the concept confused. When I am using the direct comparison test am I comparing the nth partial sums? Or am I comparing the sequence?

Answer 8

The sequence :)
Each individual term of the sequence, you compare

Answer 9

so then the 1/n! would be\[\frac{ 1 }{ 0! }+\frac{ 1 }{ 1! }+\frac{ 1 }{ 2! }\] etc?

Answer 10

Yes :D

Answer 11

and I need to find a sequence that diverges that is less than this sequence or one that converges that is greater than this sequence?

Answer 12

Yup.

Answer 13

Yikes!!

Answer 14

It doesn't have to be greater all the time. Just that at some point, it will always be greater/smaller

Answer 15

right but I thought factorials grew faster than exponentials?

Answer 16

In this case, we could use this (convergent) geometric series
\[\large \frac1{2^n}\]

Now, for n = 0,1,2, and 3
\[\Large \frac1{n!}>\frac1{2^n}\]

But for n = 4 onwards
\[\large \frac1{n!}<\frac1{2^n}\]

So you can use the comparison test there ^.^

Answer 17

I am a little confused, I thought somewhere along the line someone said that factorials grew faster than exponentials.

Answer 18

of course, that's why when factorials are in the DENOMINATOR, they become smaller than when you have EXPONENTIALS at your denominator :)

Answer 19

Duh LOL I was just testing you see if you remembered lol

Answer 20

Is that so? -.-

Answer 21

Thanks that helps. I was just having a slow moment