Visualizing Limits with Factorials and Powers

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sedaw
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how to calculate the limits ?

resize.asp?image=1_124518892.jpg


appreciate any kind of help ...
 
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n!/n^n

2^n/n!

n!/2^n

1/n + [1 + (-1)^n] / [2^(1/n)]
 


You should show that you have done a little work on the problems.

Here is a hint to visualize some of the limits. n! is the product of n terms, one of them is n and the rest are all less than n. n^n is the product of n terms, all of them are n.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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