Uni Maths Questions

[vcestudent94]

Can someone help me with this sequence limit question?
Lim n-->infinity (3^n - 4^n)/(3*n^2+4^n+7)

[BubbleWrapMan]

divide numerator and denominator by 4^n

[vcestudent94]

Thanks. How about this sequence:
(3^n+2!)/(5^n-n!)

And also, how would you use the comparison test to determine if sums like this one is convergent or not?
sum( (sqrt(n) - 1)/(n^2 + 1) )

[vcestudent94]

Bump

[Nagisa]

And also, how would you use the comparison test to determine if sums like this one is convergent or not?
sum( (sqrt(n) - 1)/(n^2 + 1) )

yes, this is how i would do it



the comparison test states that if you have two series (say, and ). if is convergent and then is also convergent.
conversely, if diverges and then is also divergent.

so this means in this case we can look at the dominant parts of the series which are on top and on bottom.

so we should choose our to be

now we have             (since now we have as a  'p series' (1/[p^n]). a p series converges if n > 1 and diverges if n < 1)

so converges and via the comparison test, so does

[vcestudent94]

Thanks! I get it now.. would you happen to know how to approach this question:

Lim n->infinity [(3^n+2!)/(5^n-n!)]

[Nagisa]

do you mean show convergence or divergence?

[kamil9876]

I'm wondering if there is a typo in there since why write instead of just        ?

Anyhow, assuming this is what you meant, then divide top and bottom by 3^n and you see that you get .

The numerator goes to and so if you can show the denominator then you have shown that it goes to . It isn't hard to see that if and then grows much faster than and the point is that while so from this you can show that eventually will be larger than and from then on will grow much faster and hence the difference goes to infinity. (As an exercise you can turn this into a more precise argument).

[vcestudent94]

Yes its a typo. Sorry, I meant 2n! instead of just 2!

[kamil9876]

Just before I solve an irrelevant problem again, is that (2n)! or n! ? In either case you should multiply by and then use the fact that for each fixed . So the limit is either or depending on whether you meant or respectively

[nubs]

How would you use the sandwich theorem to find the limit of (3^n+1)^(1/n) and n!/(n^n)?

[kamil9876]

[vcestudent94]

Find the 576th derivative of e^(4t)*cos(4t) ?

[BubbleWrapMan]

Express it as

[Jeggz]

 

 





Then just differentiate from there, hope that helped 

EDIT: Beaten