Question

Determine whether the sequence converges or diverges. How do you know if it converges or diverges? If it converges, give the limit.{ 5n−1 / n+1 }

Answer 1

Given the sequence,

`((5n-1)/(n+1))`

We can find the solution to the question below.

Step-by-step explanation

1) The sequence converges.

2) This is because the limit of the sequence exists as n→∞.

We can find the limit below.

`\begin{gathered} \lim_(n\to\infty\:)\left((5n-1)/(n+1)\right) \\ divide\text{ the numerator and denominator by n} \\ \lim_(n\to\infty\:)\left((5-(1)/(n))/(1+(1)/(n))\right) \\ Recall;\lim_(x\to a)\left[(f\left(x\right))/(g\left(x\right))\right]=(\lim_(x\to a)f\left(x\right))/(\lim_(x\to a)g\left(x\right)),\:\quad\lim_(x\to a)g\left(x\right)0 \\ (\lim_(n\to\infty\:)\left(5-(1)/(x)\right))/(\lim_(n\to\infty\:)\left(1+(1)/(x)\right))=(5)/(1)=5 \end{gathered}`

Answer: The limit of the sequence is 5