finding an 'e' based limit for this sequence

Question

i need to find the limit for

$$\lim_{n \rightarrow \infty} \left(1 + \frac{q}{n}\right)^n $$

where $q \in \mathbb{Q}$

how to i get this sequence to resemble

$$\lim_{n \rightarrow \infty} \left(1 + \frac{1}{n}\right)^n $$

so i can find its limit?

Answer 1

$q = \frac{s}{t}$, where $s,t\in\mathbb{N}$, so

We can write $$\left(1 + \frac{q}{n}\right)^n = \sqrt[t]{\left(1 + \frac{s}{tn}\right)^{nt}}$$,

for the term inside the bracket, i.e.

$$\left(1 + \frac{s}{tn}\right)=\frac{tn+s}{tn}=\frac{tn+1}{tn}\cdot\frac{tn+2}{tn+1}...\frac{tn+s}{tn+s-1}=\left(1+\frac{1}{tn}\right)\cdot\left(1+\frac{1}{tn+1}\right)...\left(1+\frac{1}{tn+s-1}\right)$$

Now it should be clear what you should do. Let me know if you still have problems.