How do you determine whether the sequence $a_n=n-n^2/(n+1)$ converges, if so how do you find the limit?

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Answer 1 · 2017-07-27T19:12:08

The given Seq. converges to $1.$

Observe that, $a_n=n-n^2/(n+1), (n in NN)$

$={n(n+1)-n^2}/(n+1),$

$rArr a_n=n/(n+1).$

We know that, as $n to oo, 1/n to 0.............(ast).$

$" Therefore, the Reqd. Sequencial Limit="lim_(n to oo) a_n,$

$=lim_(n to oo) n/(n+1),$

$=lim_(n to oo) canceln/{canceln(1+1/n)},$

$=lim_(n to oo) 1/(1+1/n),$

$= 1/(1+0),................[because, (ast).]$

$" The Reqd. Sequencial Limit="1.$

Because, the Seq. Lim. exists, the seq. converges to $1.$

How do you determine whether the sequence a_n=n-n^2/(n+1)a_n=n-n^2/(n+1) converges, if so how do you find the limit?