Help What series get larger value when n is sufficiently large

In summary, The conversation discusses which series, n! or n^(10^10), gets larger values when n is sufficiently large. The use of Stirling's approximation is suggested but not necessary for solving the problem. It is mentioned that a value of n exists where n/2 is larger than 10^10 and that bounding n! from below by a polynomial for very large n is a possible approach. However, using Stirling's approximation may not be allowed for the coursework.
  • #1
esvee
5
0

Homework Statement



i'm stuck on this question for a long time now, any help would be greatly appreciated..

which of the series gets larger values when n is sufficiently large:

n! or n^(10^10)
 
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  • #3
You don't really need that. There is a value of n so that n/2>1010 (obviously)

Now try to bound n! from below by a polynomial for very large n
 
  • #4
Office_Shredder said:
You don't really need that.

Obviously you don't, it just makes the problem considerably simpler.
 
  • #5
Office_Shredder said:
You don't really need that. There is a value of n so that n/2>1010 (obviously)

Now try to bound n! from below by a polynomial for very large n

thanks for the replies!

i still don't get it, i tried various ways using the sterling approximation (i'm pretty sure I'm not allowed to use it in this coursework) and by trying to take log10 out of both of the "inequality's" sides... i can't find use for the fact that n/2 is larger than 1010.

*going crazy*
 
  • #6
jgens said:
Obviously you don't, it just makes the problem considerably simpler.

The problem with using Stirling is that you wouldn't have rigorously proved the statement if you don't know how to derive Stirling rigorously (including with a rigorous error term).
 

FAQ: Help What series get larger value when n is sufficiently large

How do you determine if a series will have a larger value when n is sufficiently large?

To determine if a series will have a larger value when n is sufficiently large, you can use the ratio test or the comparison test. These tests involve taking the limit of the series and comparing it to a known convergent or divergent series. If the limit is greater than 1, the series will diverge and have a larger value. If the limit is less than 1, the series will converge and have a smaller value.

Can a series have a larger value when n is sufficiently large even if it is a convergent series?

No, a series cannot have a larger value when n is sufficiently large if it is a convergent series. This is because a convergent series means that the sum of all the terms in the series approaches a finite value as n approaches infinity. Therefore, the value of the series cannot get arbitrarily large as n increases.

What factors can affect the value of a series when n is sufficiently large?

The value of a series when n is sufficiently large can be affected by the rate of growth of the terms in the series and the type of series it is (e.g. geometric, harmonic, etc.). Additionally, the convergence or divergence of the series can also impact its value when n is large.

How does the behavior of a series change when n is sufficiently large?

When n is sufficiently large, the behavior of a series can change from convergent to divergent, or vice versa. This is because as n increases, the terms in the series may grow or shrink at a certain rate, causing the overall behavior of the series to change.

Are there any other methods for determining if a series will have a larger value when n is sufficiently large?

Yes, there are other methods for determining the behavior of a series when n is sufficiently large. Some additional tests that can be used include the integral test, the root test, and the alternating series test. These tests involve comparing the series to an integral, a root expression, or an alternating series to determine its convergence or divergence.

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