Show Limit Theorem: Sum of Sequence is L

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Homework Statement


Suppose that a_n->L as n->infinity. Show that (a1+a2+...+an)/n=L as well.


Homework Equations





The Attempt at a Solution


I'm thinking something about limit theorems here?
 
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use the definition. note that |(a1+a2+...+an)/n-L| = |((a1-L)+...+(an-L))/n|, and I guess you know something about the behavior of |a_n-L| when n goes to infinity :)
hope this helps u
 
Sheesh, should have seen that. Thanks!
 
economist1985 said:
Sheesh, should have seen that. Thanks!

No, you shouldn't have seen that. That's not a proof at all. You need to go back to epsilons and deltas for this one.
 

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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