Calculus Answers

Determine if the following series converges?
Infiniti
Series (n!)^2/(2n)!
n=1

Trace the curve xy^2=4a^2(2a-x)

1)Find lim x->0 (1/x-1/e^x-1) using L`hopital`s rule.
2)Using integration by part, evaluate
intgrate from the limit square root of e to e of ln x / x^2 dx
3)using appropriate substitution evaluate ingration of -1 to 1 of ydy / square root (y+1)

If F(x)〓x.x and G(x)〓|x|, then both FoG and GoF are differentiable at x〓0 being equal to x.x (or x squared). But G(x) is not differentiable at x〓0. Does this contradict the chain rule requirements where both the functions must be differentiable at their respective domain points?

Sir,
I'm having a concept problem and would be glad if you can help me out.
my problem is as follows -
1. l dx = x + c right? What does this actually mean? does it mean we are integrating a very small x over and over so we get the parent function?

2. l [v(0), v(t)] dv = v(t) - v(0) ,how? here v is basically a f(t) ,thus to integrate, doesn't it mean
l[0,t] df(t) = f(t) - f(0)?

3. | dv = v, not v+ c, why? where v is velocity
please It would help me a lot if you can clear my concept.

Find all the solutions of 6cos x - 3√3 = 0

Find all the solutions in the interval [0,2π) for the equation cos² x - 2cosx - 3 = 0
Show your Work.

dear expert give an example of comutative law in real world.Best wishes

I need to develop a power series of

(a)
ln (1+x / 1-x)

(b)
Using the result, calculate with the first four exact decimals the number ln 2.

i want to know how i can fiend ( lim ( ln(cosx)/ln(cos3x)) when x->0 i know answer is (1/9)
but i dont want use L hospital rul