[Potironette]

So..infinity is a number? What does "closed form" mean?



For math homework, there's a problem that says to find the ratio, fifth term, and the nth term of a sequence: 5, 5^c+1, 5^2c+1, 5^3c+1, ...

I got that a₅ = 5^4c+1 and that a_n = 5^(n-1)c+1.
But what is the ratio? Is it just 5 or is it 5^(n-1)c+1 or something else?



Also, how does S_n = (1/2)(n(2a +(n-1)d)) = n((a₁+a₂)/2) ?
(That is, how does (1/2)(n(2a +(n-1)d)) turn into n((a₁+a₂)/2) ?)
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How does it work that the magnetic field lines points north inside a magnet? What really is a magnetic field? If I remember right it's something about where the north point of the compass would point but..that means if a compass were somehow stuck inside a magnet it would point north. What's actually going on..?

[Coda]

Infinity isn't EXACTLY a number but you can use it like one in many cases.

I've explained "closed form" before. Basically it's a function that returns the same values as a sequence if you pass the natural numbers to it, but it doesn't have any recursion (that is, self-reference) in it.

The ratio is just S_n / S_n-1. For a sequence like this one the ratio is constant across any two consecutive terms. Spoilering the answer:

5^c+1, because a^b / a^c = a^b-c

Without context, I have no idea on Sn = (1/2)(n(2a +(n-1)d)) = n((a1+a2)/2), because I don't know what a, a1, a2, and d are.

What really is a magnetic field?

I mean, you've been studying fields. A magnetic field is a polarized field, just like an electric field. They behave the same way, they're just based on different (but related) underlying forces. "North" in a magnetic field and "positive" in an electric field are analogous.

The compass needle always aligns itself along the field lines, because one end is attracted to the north pole of the magnet and the other end is attracted to the south pole of the magnet. (Of course, that means the pointing end of the compass needle is in fact the compass's SOUTH pole. Don't let that confuse you.)

[Potironette]

(Of course, that means the pointing end of the compass needle is in fact the compass's SOUTH pole. Don't let that confuse you.)

I thought the pointing end was the compass's north pole..?

I've explained "closed form" before. Basically it's a function that returns the same values as a sequence if you pass the natural numbers to it, but it doesn't have any recursion (that is, self-reference) in it.

Oops, I forgot ._.'''. So basically I can find the sum at any position (n) with closed form and that's what closed form is?

5c+1, because ab / ac = ab-c

So to find a ratio I just look at one value and the next and if I don't know how much to multiply the first by, I just divide the next from the previous! Although, wouldn't that be 5^c?

Without context, I have no idea on Sn = (1/2)(n(2a +(n-1)d)) = n((a1+a2)/2), because I don't know what a, a1, a2, and d are.

I messed up writing it, woops. a = a₁. And 'a' represents the first term in an arithmetic series. d is the difference between two terms in an arithmetic series. S_n in this case is the sum for the arithmetic series. I learned to use those formulas to find the sum of things such as 1, 3, 5, ... up to a specified nth term in class.

[Coda]

Originally Posted by Potironette

I thought the pointing end was the compass's north pole..?

Nope, because opposite poles attract. It's the part of the compass LABELED north, but magnetically speaking it's the south pole.

Oops, I forgot ._.'''. So basically I can find the sum at any position (n) with closed form and that's what closed form is?

Yup.

So to find a ratio I just look at one value and the next and if I don't know how much to multiply the first by, I just divide the next from the previous!

Yup.

Although, wouldn't that be 5^c?

Um...

5^nc+1 / 5^(n-1)c+1
5^{(nc+1) - ((n-1)c+1)}
5^{nc + 1 + -(nc + -c + 1)}
5^{nc + 1 + -nc + c + -1}
5^{nc + -nc + 1 + -1 + c}
5^c

... yep, you're right, my mistake!

I messed up writing it, woops. a = a₁. And 'a' represents the first term in an arithmetic series. d is the difference between two terms in an arithmetic series. S_n in this case is the sum for the arithmetic series. I learned to use those formulas to find the sum of things such as 1, 3, 5, ... up to a specified nth term in class.

That means that d = a₂ - a₁, so you can substitute that into the expression.

n(2a₁ + (n-1)d)/2 = n((a₁+a₂)/2)
n(2a₁ + (n-1)(a₂ - a₁))/2 = n((a₁+a₂)/2)
2a₁ + (n-1)(a₂ - a₁) = a₁+a₂
2a₁ + (n-1)a₂ - (n-1)a₁ = a₁+a₂
2a₁ - (n-1)a₁ + (n-1)a₂ = a₁+a₂
(2 - n - 1)a₁ + (n-1)a₂ = a₁+a₂
(1 - n)a₁ + (n - 1)a₂ = a₁+a₂
(n - 1)(a₂ - a₁) = a₁+a₂

... by all accounts it doesn't make sense. Did I make a mistake, or did you transcribe something else wrong?

[Potironette]

...Woops. It's actually
n(2a₁ + (n-1)d)/2 = n((a₁+a_n)/2)

2a₁ + a₂(n-1) - a₁(n-1) = a₁ + a_n
a₁ - a₁(n-1) + a₂(n-1) = a_n
a₁ - a₁n + a₁ + a₂n - a₂ = a_n
2a₁ - a₁n - a₂ + a₂n = a_n
a₁(2-n) - a₂(1-n) = a_n

It works now, but I'm not sure what it's saying.


Err do you mean the north of the earth is labeled north but it's actually magnetically south?

[Coda]

That is... entirely possible. I don't actually know. If the "north" side of the compass needle points to the south pole of a permanent magnet, then you are correct.

What it's saying is if you know the first term of a sequence, and if you know the (constant) distance between subsequent terms, then you can find any term of the sequence using that last line you derived, and you can find the sum of the sequence at any point using the identity you posted.

It should be noted that this follows from theorems you already know:

Σx = n(n+1)/2 -- the sum of the natural numbers
a(b+c) = ab + ac -- the distributive property of multiplication over addition
a + a + ... + a = na -- the additive definition of multiplication

Why?

Well, consider the sum of the integers:
1 + 2 + ... + n = n(n+1)/2

Our life is going to be much easier if we start with zero, though, so let's substitute (n-1) in place of n:
0 + 1 + ... + (n-1) = (n-1)n/2

Multiply it by d:
d(0 + 1 + ... + n) = d(n-1)n/2

Distribute the d:
0 + d + 2d + ... + nd = d(n-1)n/2

Add a₁ to each term, which is the same as adding n copies of a₁:
a₁ + (a₁+d) + (a₁+2d) + ... + (a₁+nd) = d(n-1)n/2 + n(a₁)

But of course, a₁+d is a₂:
a₁ + a₂ + ... + a_n = d(n-1)n/2 + n(a₁)

[Espy]

Coda suggested I put this here for easier viewing :P

This problem has to do with Gauss Quadrature, specifically one-point to five-point Gauss-Legendre formulas.

Here's the problem:

Write a function which computes the integral of a function handle f using the Gauss-Legendre quadrature rule. The function takes as input the function handle f to be integrated, number of Gauss-Legendre points n chosen for integration and the limits of integration a and b.

Refer the following link, to program the weights and points of Gauss-Legendre Quadrature from n = 1 until n = 5.

https://en.wikipedia.org/wiki/Gaussian_quadrature

And here's my code so far.
I've split it into two parts: the first part of the if statement deals with cases where n is even, and the second where n is odd. (n is the number of points used in the quadrature, and corresponds to the row of both matrices.)

I've tried it against a test code and an example from the text, both where n = 3, and both passed. I tested the example with n = 4, and since Gauss quadrature is accurate to the third degree, and the solutions for n = 3 and n = 4 were the same, I'm reasonably certain that the first half is okay. However, using n = 5 for that example, the answer given didn't match the actual value of the integral, or the solutions for n = 3 and n = 4.

My code fails an assertion (test), but that test is hidden, so I'm not sure which part of my code is messing up. However, since the solution for n = 5 seems wrong, I'm assuming that might be it.

[Coda]

Espy and I discussed this in chat... and failed to find an answer. :x I have been bested! ^^()

[Potironette]

As an object is thrown, does it accelerate with anything other than gravity downwards (no air resistance)? That is to say, I throw a baseball cap into the air, and once I release the baseball cap, is the baseball cap accelerating anything because of me or has that ceased?

[Coda]

The answer to your question is F=ma. If you're not touching it, F=0.

[Potironette]

Ohh I see. Acceleration needs a force.

Also, if an object were spun in a circle at a constant speed, would there be just a force to the center of the circle or would there be two forces: one to the center of the circle and another to the side?

And an English question: in the above question I used a colon then a question mark. Should I have used parentheses or does it even matter?
Another English question: Between "question mark" and "should," could I haave used a semi-colon?

[Coda]

If the object is spinning at a constant angular velocity, then the net force acting on that object is directed towards the center of rotation. If the object is speeding up, then you have that force as well as a (straight-line) force going straight forward that's causing the object to accelerate.

In practice you still have to apply a force going forward to keep an object spinning due to energy loss from air resistance, friction in the rope, etc. But the net tangential force is zero if it's spinning at a constant rate.

The colon is a fairly weak choice there. You COULD use a dash, a colon, a comma, or parentheses in that context. Each one has a slightly different nuance. Parentheses, for example, deemphasize their contents relative to the sentence, and you could take the contents out without impacting the meaning. A comma suggests that the clause that follows describes the noun before the comma. A colon suggests that the part afterward specifically follows from or equates to the idea introducing it. And a dash would function like parentheses except with stronger emphasis. These aren't hard and fast rules, and my interpretations of them are certainly not the only way to look at them, and you definitely shouldn't try to read too much into a statement based on its choice of punctuation.

A semicolon wouldn't have been wrong in that context but a period is better because the connection between the two thoughts isn't especially strong. Either sentence is strong on its own without need for further information.

[Potironette]

Thanks for the responses!

So...

[Potironette]

Why is theta = -pi/2 the same as x = 0 (my text-book's answer sheet)? Isn't it only the bottom half of x = 0?

[Espy]

Originally Posted by Potironette

Thanks for the responses!

So...

I don't think the orange one exists, unless you're also, say, pushing the spinning object towards the right. (Perhaps explain your reasoning behind the orange force?)

Originally Posted by Potironette

Why is theta = -pi/2 the same as x = 0 (my text-book's answer sheet)? Isn't it only the bottom half of x = 0?

In what context?

[Potironette]

Thanks for the responses!

I wasn't sure if the orange force would exist or not because I was wondering if there needed to be a force towards the side in order for a ball on a string to swing around in a circle. Apparently that's a fictitious force though..?

I don't have the sheet with me anymore, but I think the problem was: change the polar equation into a coordinate equation.
The polar equation I was given was: theta = -pi/2
(Actually, now that I think about is, is theta = -pi/2 the same as the coordinate equation x = 0 because theta = -pi/2 means that (-n, -pi/2) and (n, -pi/2) are both included?)