Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
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/… –  anorton May 7 at 22:33

1 Answer 1

up vote 2 down vote accepted

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

share|improve this answer
    
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
2  
maybe I am thinking too much haha @user1333057 –  freak_warrior Nov 16 '13 at 11:26

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.