But even the series you are using tells you the result.
Since it converges for all x, we have that for all x
(be careful with what you write, and )
I was just wondering while in the shower if the following limit
Can be done another way...Please...Please don't put down the ways that are commonly seen...I know that
increases much faster than thus the limit is 0...but does this method work?
For aesthetics n will denote ∞
Factoring out an from top and bottom we get
Cancelling we would get
Which would be equilvalent to
Is that mathematically sound?
What you have is not a rewrite of the given limit but a new limit:
This is a truncating the top and bottom of expansions of the terms in the original limit after terms so at best you have shown that the limit of the ratio of the truncations is zero.
So now how do you relate to ?
Answer that question satisfactorily and you will have a valid demonstration.
(use of one of Stirlings asymtotic formula for is nicer as it give the asymtotic form for the ratio for large )
And Would the two limits relate by or in other words by the fact that we showed that we have Thus by the squeeze theorem ?
For fixed x the largest n=x, so arbitary values for n are not available for a double limiting process.