If the string an=(n+1)/(5n+2), determine that the limit is 1/5.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the limit of the series `(a_n), n>=1` , such that:

`lim_(n->oo) (a_n) = lim_(n->oo) (n + 1)/(5n + 2) = oo/oo`

Since evaluating the limit yields an indeterminate result, you may factor out n such that:

`lim_(n->oo) (a_n) = lim_(n->oo) (n(1 + 1/n))/(n(5 + 2/n))`

Reducing duplicate factors yields:

`lim_(n->oo) (a_n) = lim_(n->oo) (1 + 1/n)/(5 + 2/n)`

`lim_(n->oo) (a_n) = (lim_(n->oo) 1 + lim_(n->oo) 1/n)/(lim_(n->oo) 5 + lim_(n->oo) 2/n)`

`lim_(n->oo) (a_n) = (1 + 0)/(5 + 0) =>lim_(n->oo) (a_n) = 1/5`

Hence, evaluating the limit `lim_(n->oo) (a_n)` yields that `lim_(n->oo) (a_n) = 1/5` , hence, the statement is valid.

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