if a_n is a convergent series ,then is a^2 _n always convergent? give example

Question

anonymous · Answer

does this ask 
"if
$$lim_{n\rightarrow \infty}a_n=L$$ then does 
$$lim_{n\rightarrow \infty}(a_n)^2=L^2$$ ?"
or does it ask 
"if 
$$\sum a_n = L$$ does 
$$\sum (a_n)^2
$$ exist?"

anonymous · Answer

$$a _{2^n}$$
I think it asks for the convergence of this. 
And the answer is yes. 

This is a good tool for slowly converging series because 2^n terms will converge much more rapidly.

anonymous · Answer

@andras you may be right but that is certainly not what is written

anonymous · Answer

It is a bit confusing what is written, I just assumed that this is it.

anonymous · Answer

says "a^2-n" which if i write in latex comes out as 
$$a^2_n$$

anonymous · Answer

but who knows?

anonymous · Answer

he wrote a^2_n ... that can be anything :-)

anonymous · Answer

why is that so?

anonymous · Answer

in which case i am trying to think of a counter example where the sum converges because the original series alternates but all my exmples turn out not to work because if 
$$\lim a_n=0$$ then 
$$\lim a^2_n $$ goes to zero fast enough

anonymous · Answer

no wait, i asked her in the chat and she said she meant the series (a_n)^2

anonymous · Answer

sorry my question is 
if $$a_n$$ is a convergent series ,then is $$\sum_{i}^{\infty}a_n^2$$always convergent? give example

anonymous · Answer

$$ t \rightarrow \infty (e ^{t^{n}})$$