@Mann @AjayMishra

@Dante where do you think it's not bounded?;

At n=1

@Mann

Oh what you are defining is not a sequence in the first place

@Dante

A sequence is a function from natural number into some set X

That means it is well defined for all natural nu.ber

It's not the case that function is unbounded but rather the function is not well defined.

@Dante I also have the same doubt

@Mann how can we redefine the sequence 1/(1-n) ?

-1/n

@AdvilSell @Dante

@Mann means ?

I think the main doubt is whether the function 1/(1-n) is bounded on the real numbers ? Or whether on it's domain or unbounded or anyother permutations

@AdvilSell

I think you are confusing definitions

Can you state the question more clearly?

@Mann leave that question

Can we say that 1/(1-n) represents a sequence ?

Or any function of form 1/(a-n) , a belongs to natural numbers , can represent a sequence ?

Or to say :

Is this ^ true ?

@AdvilSell

A symbolic formula is never a sequence

A sequence is a function from natural number to an arbitrary set. (Most generally accepted definition in standard analysis)

The domain and range of the function is a part of definition of the function.

If you are saying that f : N to R defined by the symbolic representation f(n) = 1/(1-n) is a sequence

Then it's a NO

This is not a sequence

Because the function is not well defined for n = 1

Oh,

So, 1/1-n is not a sequence at all?

Nope

@Dante

its not

Okay, cleared, thanks!

Cleared?

People still get issues after this point many a times haha

Be sure?

@Mann thanks man !

Cool @AdvilSell :D

@Mann I have something to ask [offtopic] for JEE (physics)

Say @McSuperbX1

I wanted to know how should I prepare for physics?

For JEE-A ofcourse.

I have started to solve Irodov.

Follow axiomatic pathways

And form proper causal relations

In short:

I don't understand :(

Let me devise an example

Suppose you have a sphere of finite radius. And on the surface some charge is stuck, say it's positively charged. It is rolling on an inclined surface. A magnetic field is applied perpendicular to the inclined plane. Draw me a figure and explain the motion of the sphere.

Try to explain this question to me.

Please give be a bit of time.

Ofcourse.

I think that the sphere should not go down the plane in a straight fashion

It would rather curve around, but will still go down

(Sorry for the awful handwriting, realized after I uploaded this image)

I will attempt to explain whatever I was able to think:

So I assume the charge is evenly spread on the surface.

Now at some instant while the sphere rolls, Some charge will have a velocity exactly parallel or antiparallel to the magnetic field. These do nothing.

Now some charges will be in direction of the motion and some will be in opposite direction of motion.

Actually hold on, I think the sphere rolls down like a normal uncharged sphere

^

Go on

Well then it is a regular ball rolling down an incline plane problem

Magnetic field's effect is exactly nullified

I am sorry. I was wrong in the terminology of perpendicular to plane. Say you have a 2 dimensional view in which you can see the inclined and as a line and a sphere as a ball rolling on it

The field is perpendicular to this 2D view

Oh okay.

So if you can see the motion as a ball rolling down a plane in some x y plane. Z direction is the field

Let me think about your way until

Yep got it.

Ok.

According to your way I think your first answer is correct

@Mann That is also what I think now. There is still a net velocity that is, going downhill.

You have to see the velocity from the ground frame and it's always toward right side for each charge in some sense.

And the direction you indicated is also correct

In the drawing

Now I will explain what I meant by the approach I had mentioned.

Ok.

So first, define the building block for this problem.

There are two phenomenon. Purely magnetic and Newtonian.

And they both have certain axioms, which you take as truth without questioning. It's not like they are derived from somewhere, it's just that they are.

Agreed

(they can be derived if you are interested enough but for now that's okay)

But you will always have some axioms.

Here you start with lorentz law and Newton's law.

Yes

Now here's the thing, both of these laws requires the specification of a frame of reference. So you choose that. Newtonian one will be better with inertial frame of reference. So, you take the frame fixed with inclined plane.

There's all the definition and axioms you had needed to analyze the problem.

ok.

Now you look at the object and notice how the object with it's properties will respond to these laws and constraints you have put it under. Since the question asks only qualitative picture. You can start with what happens initially

In my case : lorentz law does nothing. Either the force is symmetrically outward or symmetrically inwards

@Mann In the "Second version" of the question right?

And the Newtonian laws only says that the ball will roll down under gravity.

Yes the one I had envisioned

I will come to yours as well

Since the Newtonian laws ensures that the direction of velocity never changes or XY plane never changes we know this is the ultimate motion. Problem done

@Mann Isn't there a net force (by magnetic field) vertically above (opposite to $mg$)?

Due to the net velocity of the positive charges going down the slope

Achaa haan sorry, I am very sorry.

Yes that's the case.

Sorry, I mean net force perpendicular to the plane, not $mg$

I also made the frame mistake haha

It's not exactly verticly opposite though

Yes exactly, perpendicular to the inclined plane.

What you can say is you can come at a equilibrium velocity

How?

Hold on please.

Didn't crop (still learning FOSS software)

Ahhh okayy okayy

But you should be proving this by integrating though

@Mann Integrate? What should I integrate?

dF = dq V x B

Over the whole periphery

Oh. I simply skipped that.

Thats wrong to do.

Why?

I mean, wrong to skip.

Not wrong to integrate lol

Haha yess. But you can see with symmetry I think it will be perpendicular

Though

@Mann I dont get you. Net force by magnetic field is perpendicular to the plane right?

Since the force is still in the plane.

Yess

Yep :-)

You can imagine that it will eventually have a equilibrium velocity v down the plane.

Possibly*

Or

It can keep on accelerating downwards

@Mann Why is that? Wouldn't the magnetic force re-enforce acceleration to the right?

I am saying this because, rolling needs friction.

Enough lift can mess up things

Oh...

Again, I forgot about the sheer existence of friction.

Now you revisit and make a causal relation.

Of all the things we discussed

Causal means : assuming this and this. This happens

Just like revising a little story?

Yes but logically.

Incident 1 --> Incident 2 --> .... --> Final conclusion

Oh okay.

Yes

The links are logical statements

I often forget whatever conclusions I came, for example in this question I did apply logic and came to the correct answer

Noo there's the issue.

But after a period of time - poof everything is gone. If you were to give me the same question, I bump into the same issues - giving an incorrect answer in the first go

@Mann Exactly!!

People in your age have a big confusion with formal logic vs intuitional logic

@Mann What are the two?

These may not be technical terms so don't take it literally

But loosely speaking

Okay sure

Formal logic is water proof and tight argument. It's as strong as a mathematical proof.

@Mann Which is what we discussed above, correct?

Intuitional logic consistently works based on intuition and feelings and seems correct and logical but is not water proof.

During the whole course of such logic many assumption are taken

Yes true!

Underlying and unaware

The issue is that it's underlying and unaware

That's what causes all the problem

We did sort of Intuitional one.

For example, we haven't proved the force will always be perpendicular

Did we?

Well... we didn't prove that. yeah.

It just felt like

And this is why you miss out so much things later on

Because they once felt like

We didn't solidified or internalized them

However, when you take the time to give a formal proof of it. They will always remain.

Well mostly

I see.

This is what I meant by axioms and causal links in short.

Make a clear pathway of reasoning and know exactly what you assume

Now try this out on the above things we have done so far.

What a way to teach something! :)

Haha thanks!

Say something accurate yet complicated

"What does that mean?"

"Let me explain myself"

I think this is a good approach

builds up some curiosity

Anyway - about the book.

Yes that will be fine. But remember the main thing is you know your ground very well. Think of your reasoning like a structure of steel rods making connections like a heirarchy.

@Mann Ok!

And you are travelling across that based on a few grounds. Now, if you don't have the assumptions or grounds clearly stated. You really don't know where you are coming from or where you are going

Okayy I gotta go now though. Cya

Okay.

Thanks a bunch.

:D

@Mann cleared for now, I meant 😂