Convergence of $ x_n = \left( \frac{1}{2} + \frac{3}{n} \right)^n$ I need to show that the sequence $ x_n = \left( \frac{1}{2} + \frac{3}{n} \right)^n$ is convergent.
Using calculus in $ \mathbb{R}$, we could see that $ \lim _{n \to \infty}  \left( \frac{1}{2} + \frac{3}{n} \right)^n = \lim_{n\to \infty} e^{n \ln \left( \frac{1}{2} + \frac{3}{n} \right)} = 0   $. I do not need to calculate the limit of the sequence, just to check it convergence. The definition of convergence doesn't look like a good idea, so I've been trying to use a trick like we do in calculus or to show that the sequence is bounded and monotonic. My guess is the last idea is better than the first one. If I show that $ x_{n+1} < x_n $, since $x_n $ is clearly bigger than zero we are done. But how to show it? i've tried induction withou success. Is there a better way to show it?
Thanks in advance

Hint: $\dfrac12+\dfrac3n \le \dfrac78$, when $n\ge8$.
