# Thread: Study the convergence of a sequence

1. ## Study the convergence of a sequence

Hello all.

How do you study the convergence of a sequence?
If you can, try to exemplify with this example:
$\displaystyle U_n = 5 - \frac{2}{\sqrt[3]{n}}$

2. Have a read of this.

Ratio test - Wikipedia, the free encyclopedia

Your sequence looks like it converges to 5.

3. Originally Posted by Wright
Hello all.

How do you study the convergence of a sequence?
If you can, try to exemplify with this example:
$\displaystyle U_n = 5 - \frac{2}{\sqrt[3]{n}}$

Arithmetic of limits: $\displaystyle \frac{2}{\sqrt[3]{n}}\xrightarrow [n\to \infty] {} 0\,,\,\,and\,\,5\xrightarrow [n\to \infty] {} 5$, so the limit of both summands exists and finite and thus the limit is the sum of these two limits, i.e. 5.

Tonio

4. Originally Posted by pickslides

Ratio test - Wikipedia, the free encyclopedia

Your sequence looks like it converges to 5.
Be careful.The ratio is test can be used only for series.

The only way you can relate the two is that if you show

$\displaystyle \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|< 1$

you know the series $\displaystyle \sum_{n\in\mathbb{N}}a_n$ converges which means that $\displaystyle \lim_{n\to\infty}a_n=0$ But that is not applicable here

5. So it's convergent if theres a limit and divergent if there isn't one?

6. Correct.

7. ## Limited sequence?

Mistake post.