Irrational sequence that converges to a rational limit

cnwilson2010
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Hi. I found some rational sequences that converge to irrational limits, but am not having any luck going the other direction, i.e., an irrational sequence that converges to a rational limit. Any suggestions?
 
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Can you find an irrational sequence that converges to 0? Something like \frac{1}{n} but with an irrational number in the numerator?
 
Thank you.

How about 1/n(2(1/2))?
 
cnwilson2010 said:
Thank you.

How about 1/n(2(1/2))?

Yes, that sounds good! :smile:
 
You can also use a commonly known irrational number (e.g. pi or e) and do something similar.
 
Didn't want to be oddly transcendental or anything, so I was trying to stay away from those, but found out later they would have been fine. So thank you for that.
 
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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