Suppose, $a_n$ is monotone and converges and $b_n=(a_n)^2$ . Does $b_n$ necessarily converge?

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Suppose, $a_n$ is monotone and converges and $b_n=(a_n)^2$ . Does $b_n$ necessarily converge?

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Answer 1 · 2016-01-16T15:36:13

Yes.

Explanation:

Let $l = lim_(n->+oo)a_n$ .
$a_n$ is monotone so $b_n$ will be monotone as well, and $lim_(n->+oo)b_n = lim_(n->+oo)(a_n)^2 = (lim_(n->+oo)(a_n))^2 = l^2$ .

It's like with functions : if $f$ and $g$ have a finite limit at $a$ , then the product $f.g$ will have a limit at $a$ .

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Suppose, $a_n$ is monotone and converges and $b_n=(a_n)^2$ . Does $b_n$ necessarily converge?

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Explanation:

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Suppose, $a_n$ is monotone and converges and $b_n=(a_n)^2$ . Does $b_n$ necessarily converge?