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Mathematics 31 Online
OpenStudy (anonymous):

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

OpenStudy (anonymous):

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?"

OpenStudy (anonymous):

$a _{2^n}$ 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.

OpenStudy (anonymous):

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

OpenStudy (anonymous):

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

OpenStudy (anonymous):

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

OpenStudy (anonymous):

but who knows?

OpenStudy (anonymous):

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

OpenStudy (anonymous):

why is that so?

OpenStudy (anonymous):

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

OpenStudy (anonymous):

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

OpenStudy (anonymous):

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

OpenStudy (anonymous):

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

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