# finding an 'e' based limit for this sequence

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?

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In case anyone stumbles upon this question again: There's a much easier way than the one below, see here: math.stackexchange.com/questions/568268/… – apnorton May 7 '14 at 22:33

$q = \frac{s}{t}$, 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.

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this turned out alot more complex then i think it should be, i found this thread gives a simpler answer, thanks for your help! math.stackexchange.com/questions/568268/… – guynaa Nov 16 '13 at 11:24
maybe I am thinking too much haha @user1333057 – freak_warrior Nov 16 '13 at 11:26