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.#