Closed-Form Solution for f(n) w/o Generating Fns

In summary, the conversation discusses the function f(n) and the difficulty in finding a closed form for it. The function has two different forms depending on the value of n, but it cannot be simplified further. The speaker expresses frustration with their inability to find a closed form and asks for an explanation, but then realizes they do not need one.
  • #1
Tony11235
255
0
[tex] f(n) = 0, n \leq 2 [/tex]

[tex] f(n) = \sqrt{n}f(\sqrt{n}) + n, n > 2 [/tex]

How can I get this in closed form? Generating functions won't work. Recuring a number of times hasn't worked out for me. Or can I show that f = O(n*lg(lg(n))), where lg stands for ln(n)/ln(2), without f being in closed form? Sorry for the poor tex skills.
 
Last edited:
Physics news on Phys.org
  • #2
I don't think it can be put in closed form.

Daniel.
 
  • #3
f(n)= kn for [itex]2^{k}< n \le 2^{k+1}[/itex]
 
  • #4
HallsofIvy said:
f(n)= kn for [itex]2^{k}< n \le 2^{k+1}[/itex]

Are you serious? This is the closed form I've been struggling all day to find? Could you mention how you arrived at this? Sorry it's not totally obvious to me, I'm really tired.
 
  • #5
Nevermind. No explanation needed. I'm just really pissed with myself.
 

What is a "Closed-Form Solution"?

A closed-form solution is a mathematical expression that gives the exact solution to a problem, typically involving a finite number of mathematical operations. It is often contrasted with numerical or iterative methods, which approximate the solution through repeated calculations.

What is f(n) in this context?

f(n) refers to a function that is dependent on a variable, n. This means that the output of the function, or its value, will change depending on the value of n. In the context of generating functions, f(n) is often used to represent the number of ways to obtain a certain outcome.

What is the significance of a Closed-Form Solution for f(n) without Generating Functions?

A closed-form solution for f(n) without generating functions allows for a more direct and efficient method of solving a problem. It avoids the use of generating functions, which can be complex and time-consuming to calculate. Additionally, a closed-form solution can provide a more precise and exact result compared to an approximation from a generating function.

Can a Closed-Form Solution for f(n) be applied to any problem?

No, a closed-form solution is not always possible for every problem. It depends on the complexity and nature of the problem. In some cases, a closed-form solution may be too difficult to derive or may not exist at all. In these situations, other methods such as numerical or iterative approaches may be necessary.

How can a Closed-Form Solution for f(n) be derived?

A closed-form solution for f(n) can be derived through various techniques such as algebraic manipulation, calculus, and combinatorial methods. It may require a deep understanding of mathematical concepts and techniques, as well as creativity and problem-solving skills. In some cases, it may also involve trial and error to find the correct solution.

Similar threads

Function Generation, Expressing in closed forms
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits
    • Calculus and Beyond Homework Help
Replies
32
Views
2K
B Why is Big-O about how rapidly the Taylor graph approaches that of f(x)?
    • Calculus
Replies
19
Views
2K
Show GL/O/SO(n,R) form groups under Matrix Multiplication
    • Calculus and Beyond Homework Help
Replies
2
Views
2K
Number of unique prime factors of n is O(log n/(log log n))?
    • Calculus and Beyond Homework Help
Replies
10
Views
1K
Exploring the "n+1" Sequence: Seeking a Closed Form Solution
    • Calculus and Beyond Homework Help
Replies
1
Views
995
[CalcII/DiffEq] Closed form expression for f(x) which the series converges
    • Calculus and Beyond Homework Help
Replies
6
Views
3K
Closed Set Proof: Show Limit Points of A is Closed
    • Calculus and Beyond Homework Help
Replies
16
Views
2K
Challenge Math Challenge Thread (October 2023)
    • Math Proof Training and Practice
Replies
25
Views
2K
Proof of sequence convergence via the "ε-N" definition
    • Calculus and Beyond Homework Help
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top