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

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?"

\[a _{2^n}\] 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.

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

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

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

but who knows?

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

why is that so?

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

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

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

\[ t \rightarrow \infty (e ^{t^{n}})\]

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