Answer
Convergent with limit $15$
Work Step by Step
Given: $f(x)=\lim\limits_{n\to \infty} \dfrac{5}{n}[n+\dfrac{4}{n}\dfrac{n(n+1)}{2}]=\lim\limits_{n\to \infty} 5+ \lim\limits_{n\to \infty} \dfrac{10n^2+10n}{n^2}$
The sequence converges when the limit $\lim\limits_{n\to \infty} a_n$ exists and when the limit $\lim\limits_{n\to \infty} a_n$ does not exist, then sequence diverges.
Here, we have
$\lim\limits_{n\to \infty}a_n=\lim\limits_{n\to \infty} 5+ \lim\limits_{n\to \infty} \dfrac{10n^2+10n}{n^2}$
This gives:
$\lim\limits_{n\to \infty}a_n=5+\lim\limits_{n\to \infty}\dfrac{10+10/n}{1}$
or, $=5+\dfrac{\lim\limits_{n\to \infty}(10)+\lim\limits_{n\to \infty}(10/n)}{\lim\limits_{n\to \infty} (1)}$
Thus, $a_n=15$
Hence, the sequence is convergent with limit $15$
