Alan

I see.

Ok.  To what were you referring?  (I misread your earlier reply as the sum of the natural numbers!).

Hectictar,

Guest #2 is referring to a situation that occurs in the context of the analytic continuation of the Zeta function (see https://www.youtube.com/watch?v=YuIIjLr6vUA).  However, I doubt that that context is relevant for the questioner here!

As follows:

(Because this is the Rydberg equation only the positive square root makes sense)

As follows:

For question 2, note that the total mass is 2m and this is undergoing a constant acceleration due to gravity. The constant acceleration equations allow the velocity, v, at any time to be written as:  v = u + at,  where u is initial velocity, a is acceleration (due to gravity here) and t is time interval.

For question 3, note that the 700g mass is being pulled down by gravity as is the 500g mass. However, because they are on opposite sides of a pulley, there will be a net force of (0.7 - 0.5)*9.8 N pulling the 700g mass downwards and the 500g mass upwards.  Find the time it takes the 700g mass to reach the floor using s = ut + (1/2)at², where s is the distance (20cm). Find the velocity of both 700g and 500g (the same for both, of course) from v₁ = u + at.  This will act as the initial upward velocity for the 500g mass which will carry on rising against gravity, but slowing until it comes to a stop. 

Use 0 = v₁² + 2as to find the extra distance, s, travelled by the 500g mass.  Don't forget to add on the 20cm to get the total height, and to use -9.8m/s² for gravitational acceleration in this part (as the mass is travelling in the opposite direction from that in which gravity is acting).

For question 1, the following, together with Newton's third law of motion, should help:

Do it stage by stage, as follows:

Correct!

The diagram below should help: