What is the limit of x ...

Question

anonymous · Answer

$$\sin ^{2}x $$
as x approaches infinity?

experimentx · Answer

not exactly defined http://www.wolframalpha.com/input/?i=lim+x-%3E+inf+sin^2+x

anonymous · Answer

:(( So how do you know if $$\sum_{n=1}^{\infty} (\sin n/ n)^2$$ is convergent or not?

experimentx · Answer

Oh ,,, that's a tough question!!

experimentx · Answer

$$ \sum_{n=1}^{\infty} (\sin n/ n)^2 <=\int_{1}^{\infty} \frac{\sin x}{x}dx$$

experimentx · Answer

Oops sorry ... forgot the square!!

anonymous · Answer

@experimentX indeed.
@FoolForMath Do you know how to solve this?

experimentx · Answer

I think you can use this 
$$ \frac{(\sin x)^2}{x^2} < = \frac{1}{x^2}$$

experimentx · Answer

this relation holds for all x, so you can sum them up .... and prove that it converges (limit comparison), since 1/x^2 converges ...

although don't ask me to calculate value

anonymous · Answer

Yay. thanks! How about the $$\int\limits_{1}^{\infty} 1/x^2 dx$$?

experimentx · Answer

It converges!!

anonymous · Answer

Can you show me how? It converges to one but I don't know how it got there.

experimentx · Answer

http://en.wikipedia.org/wiki/Integral_test_for_convergence $$ \int_1^\infty \frac{1}{x^2} dx = some \; finite\; value$$ check it al wolfram