Question

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

Answer 1

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?"

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

but who knows?

Answer 7

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

Answer 8

why is that so?

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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