Find the limit $a$ of the sequence $(a_n)_n$

Question

I want to find the limit $a$ of the sequence $(a_n)_n$ and the smallest natural number N such that $|a_n - a| < \epsilon \forall n \geq N$

My $a_n = 5/3 - 3^n/4^n$ for all $n \in \mathbb{N},$ $\epsilon = 1/1000$

I usually would just divide $a_n$ by n to get an $a$. However, with the n to the power this is not possible.

Please give me a hint on what to do?

UPDATE

OK, I got the limit $5/3$

When I want to calculate:

$$\left| 5/3 - (3/4)^n - 5/3\right|< 1/1000$$ $$\left|-(3/4)^n \right|< 1/1000$$

Here I am stuck...

Answer 1

Hints:

$$ 3^n/4^n=\left(\frac34\right)^n $$

• What happens when you raise $(3/4)$ to higher and higher powers?
• What then do you expect the limit $a$ of $5/3-(3/4)^n$ to be?

The definition of a limit is that $a_n\to a$ if for all $\varepsilon>0$ we have $N(\varepsilon)$ such that for all $n\geqslant N(\varepsilon)$, $|a_n-a|<\varepsilon$.

• Why does this definition coincide with your intuition about limits?
• Can you use this definition to prove that your guess for the limit of $5/3-(3/4)^n$ was correct?
• In order to prove that, you should let $\varepsilon>0$ be arbitrary and explicitly find some $N(\varepsilon)$, expressed in terms of $\varepsilon$, such that $|a_n-a|<\varepsilon$ for all $n\geqslant N(\varepsilon)$. The best way to do that is to work out the value of $|a_n-a|$ for arbitrary $n$. What can you say about the values $|a_n-a|$?
• If $n=1$, then is it the case that $|a_n-a|<\frac1{1000}$? If not, how large does $n$ have to be for that to be the case?