Limit with n tending to infinity

In summary, the function f(n) tends to 1 as n approaches infinity, provided that g(n) is also asymptotic. If g(n) is not asymptotic, then f(n) may diverge as g(n) does.
  • #1
eljose
492
0
let,s suppose that we have the limit with n tending to infinity:
[tex]\frac{f(n)}{g(n)}=1 [/tex] then i suppose that for n tending to infinity we should get:
[tex]f(n)\rightarrow{g(n)}[/tex] or what is the same the function f(n) diverges as g(n) as an special case:
[tex]\pi(n)\rightarrow{n/ln(n)} [/tex] where Pi is the prime number counting function in number theory
 
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  • #2
eljose said:
let,s suppose that we have the limit with n tending to infinity:
[tex]\frac{f(n)}{g(n)}=1 [/tex] then i suppose that for n tending to infinity we should get:
[tex]f(n)\rightarrow{g(n)}[/tex] or what is the same the function f(n) diverges as g(n) as an special case:
[tex]\pi(n)\rightarrow{n/ln(n)} [/tex] where Pi is the prime number counting function in number theory
What is your question? What you;'ve written doesn't make sense.

Are you attempting to ask: if f and g are asymptotic then does f-g tend to zero? Of course not: we can make two functions that are asymptotic diverge absolutely as fast as you want.
 
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  • #3
As I recall, that [itex]\frac{f(n)}{g(n)}=1\mbox{ as }n\rightarrow\infty [/itex] speaks of the asymptotic relationship between f and g, namely that [itex]f(n) \sim g(n) [/itex] (ref. Asymptotic Notation).
 
  • #4
My main question is related with proving (if true) the equality

[tex]\frac{\int_0^{n}dx(x^p)}{1^p+2^p+3^p+...+n^p}\rightarrow{1} [/tex]

as n tends to infinity [tex]n\rightarrow{\infty} [/tex]
 
  • #5
It certainly tends to a constant (for p an integer), as anyone can tell you, since the sum of the first p powers is a poly of degre p+1. Surely you can actually solve it, it's straightforward (especially from someone who has solved the RH amongst other things that snobbish mathematicians wont' acknowledge (you cry wolf and what do you expect?)), at least try simplifying the integral (ie doing it).
 
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1. What is a limit with n tending to infinity?

A limit with n tending to infinity is a mathematical concept that describes the behavior of a function as the input values approach infinity. It represents the value that the function is approaching as the input values become larger and larger.

2. How is a limit with n tending to infinity calculated?

A limit with n tending to infinity is calculated by first plugging in larger and larger values for n into the function and observing the resulting output values. If the output values appear to be approaching a specific value, that value is considered the limit with n tending to infinity. Alternatively, algebraic manipulation and the use of limit laws can also be used to calculate this limit.

3. What is the significance of a limit with n tending to infinity?

A limit with n tending to infinity is significant because it allows us to understand the behavior of a function at its extreme values. It can help us determine the end behavior of a function and make predictions about its behavior at extremely large input values.

4. Can a limit with n tending to infinity be infinite?

Yes, a limit with n tending to infinity can be infinite. This means that as the input values become larger and larger, the output values of the function also become larger and larger without bound. This is known as an unbounded limit.

5. How is a limit with n tending to infinity related to asymptotes?

A limit with n tending to infinity is closely related to asymptotes. Asymptotes describe the behavior of a function at its extreme values and can be horizontal, vertical, or oblique. A limit with n tending to infinity can help us identify the presence and type of asymptote in a function.

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