Answer
Limit: $2$
Work Step by Step
We are given the sequence:
$a_n=2+\dfrac{(-1)^n}{n}$
Determine the first 10 terms:
$a_1=2+\dfrac{(-1)^1}{1}=1$
$a_2=2+\dfrac{(-1)^2}{2}=2.5$
$a_3=2+\dfrac{(-1)^3}{3}\approx 1.66667$
$a_4=2+\dfrac{(-1)^4}{4}=2.25$
$a_5=2+\dfrac{(-1)^5}{5}=1.8$
$a_6=2+\dfrac{(-1)^6}{6}=\approx 2.16667$
$a_7=2+\dfrac{(-1)^7}{7}\approx 1.85714$
$a_8=2+\dfrac{(-1)^8}{8}=2.125$
$a_9=2+\dfrac{(-1)^9}{9}\approx 1.88889$
$a_{10}=2+\dfrac{(-1)^{10}}{10}=2.1$
The sequence appears to have a limit: $2$.
Calculate the limit:
$\displaystyle{\lim_{n \to \infty}} \left(2+\dfrac{(-1)^n}{n}\right)$
$=2+\displaystyle{\lim_{n \to \infty}} \dfrac{(-1)^n}{n}=2+0=2$
Plot the first 10 terms of the sequence:
