Answer
The sequence diverges
Work Step by Step
We are given the sequence:
$a_n=1+\dfrac{10^n}{9^n}$.
Determine the first 10 terms:
$a_1=1+\dfrac{10^1}{9^1}\approx 2.11111$
$a_2=1+\dfrac{10^2}{9^2}\approx 2.23457$
$a_3=1+\dfrac{10^3}{9^3}\approx 2.37174$
$a_4=1+\dfrac{10^4}{9^4}\approx 2.52416$
$a_5=1+\dfrac{10^5}{9^5}\approx 2.69351$
$a_6=1+\dfrac{10^6}{9^6}\approx 2.88168$
$a_7=1+\dfrac{10^7}{9^7}\approx 3.09075$
$a_8=1+\dfrac{10^8}{9^8}\approx 3.32306$
$a_9=1+\dfrac{10^9}{9^9}\approx 3.58117$
$a_{10}=1+\dfrac{10^{10}}{9^{10}}=3.86797$
The sequence doesn't appear to have a limit.
$a_n=1+\dfrac{10^n}{9^n}=1+\left(\dfrac{10}{9}\right)^n$
As $\left(\dfrac{10}{9}\right)^n$ is increasing when $n$ is increasing, $a_n$ also increases, so we have:
$\displaystyle{\lim_{n \to \infty}} \left[1+\left(\dfrac{10}{9}\right)^n\right]=1+\displaystyle{\lim_{n \to \infty}}\left(\dfrac{10}{9}\right)^n$
$=1+\infty=\infty$
Plot the first 10 terms of the sequence: