Recent content by lswtech

oh yes I understand your question just now. The pic you attached from Wiki is an example of simple sequences only. Pay attention to the fact that, a series is also a sequence. The same rule applys. Technically, when we are talking about "convergence", then we need to have some real L that lim...

A bounded sequence need not be convergent : (-1)^n is an example - it is bounded within 1 and -1 ( abs((-1)^n))<1 ) while it is not convergent. Series is just summation of sequences, say , Sn = 1+1/2^2+1/3^2+...+1/n^2 . Say, in fact, lim n-> infinity Sn = pi^2/6. When we say something to be...

Thank you very much! but are there any simple proofs of x>sinx and x<tanx?

my concern is - how can we show the area of sector is 1/2*pheta? and would there be any simple ways with epsilon-delta?

Oh thank you! But how to show that the Taylor expansion does equal to sinx without using sinx/x->1 at all? I am teaching a group of students which I want to make sure everything goes in the right path and get rid of "circular proofs"! thx!

(-1)^n

Yes thanks for replies above. But in fact I have watched somewhere else, saying that the area of circle = pi * r^2 is dependent on the result sinx/x = 1 as x->0 Say, cut the circle of radius r into n equal partitions (each with angle n/2pi in the center) and area of the circle = n*r^2 /2...

I would like to know if there's any proof of "limit of (sinx)/x as x approaches 0 is 1." ..

You should try setting up equations for both vertical and horizontal motions. Then you should have an unknown "d" for distance in the equations. Join x-y by cancelling the parameter "time t". Consider d as a real root and you are done!

I believe that if you connect the Battery-Resistor box with some dc supply, you will get a constant pd. only. On the other hand, an exponentially increasing potential differences, which attains a constant in the long run, will result if your box is the capacitor one.

No. Reverse (Weight pulls down the block and it is the origin of the acceleration downward)

The question stated " make the mass reverse directions in exactly 1 sec" and I thought the retardation should be 2ms^-2 but not 4. and the 2m means m*a and putting a = 2 Since the direction of retardation is pointing towards the ground, and the Net force in this direction is Friction + Weight...

Good simulations. I think I got your point now. If you have a small sphere, with velocity v, hitting a rod with the same mass at one end elastically, the result is not the intuitive case-- "the sphere stops and the rod moves forward with velocity v" but something more complicated. You need to...

Yes you are correct. It should have translation also. I was thinking about fixed axis motions only. It should be restated that angular momentum about CM as well as linear momentum are both conserved

In fact, Rotational energy is derived from Linear translation energy. Think of adding all 1/2mv^2 in the rigid body (those closest to CM have 0J, furthest particles carry max. KE) and you could get the formula 1/2 I w^2. So he's talking about that, at a single position, the particle on the...