B What is a closed form solution?

AI Thread Summary
A closed form solution is an expression that can be computed in a finite number of standard operations, such as addition, multiplication, and exponentiation. An example of a closed form solution is the finite sum $$\sum_{k=1}^5 \frac{1}{2^k}$$, which can be evaluated directly. In contrast, the infinite series $$\sum_{k=1}^\infty \frac{1}{2^k}$$ converges to a limit but is not considered a closed form solution because it requires an understanding of limits and convergence. Closed form solutions are often preferred in mathematics for their simplicity and ease of use. Understanding the distinction between closed form and open form solutions is essential for mathematical problem-solving.
docnet
Messages
796
Reaction score
488
TL;DR Summary
What is the definition of a closed form solution in math? Where did the term originate? When is it preferred?
Hi friends, I was wondering if you could give the definition of 'closed form', with examples of closed form solutions and open? form solutions.

Foe example, is this a closed form solution?
$$\sum_{k=1}^\infty \frac{1}{2^k}$$

Or this?
$$\sum_{k=1}^5 \frac{1}{2^k}$$

Thanks.
 
Last edited:
Mathematics news on Phys.org
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
Thread 'Imaginary Pythagoras'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
Back
Top