- #1
alexmahone
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Find $\displaystyle\lim_{n\to\infty}\frac{\sqrt[n]{n}}{\sqrt[n+1]{n+1}}$.
Evgeny.Makarov said:$\sqrt[n]{n}=e^{\frac{\ln n}{n}}$ and $\frac{\ln n}{n}\to 0$ as $n\to\infty$.
Do you know this limitAlexmahone said:Find $\displaystyle\lim_{n\to\infty}\frac{\sqrt[n]{n}}{\sqrt[n+1]{n+1}}$.
Plato said:Do you know this limit
$\displaystyle\lim _{u \to \infty } \sqrt{u}=~?$
What is the answer to the OP?Alexmahone said:It's 1.
Plato said:What is the answer to the OP?
A sequence is a list of numbers that follow a specific pattern or rule. It can be finite or infinite.
The limit of a sequence is the value that the terms of the sequence approach as the index (or position) of the terms increases without bound. In other words, the limit is the value that the sequence is "approaching."
The limit of a sequence can be calculated by finding the values of the terms as the index increases and observing the pattern of these values. If the values of the terms become closer and closer to a specific number, then that number is the limit of the sequence.
Finding the limit of a sequence can help us understand the behavior of a function or a real-life phenomenon. It can also help us make predictions about the future values of the sequence.
Yes, there are other methods such as using the squeeze theorem, the ratio test, or the root test. These methods are especially useful for finding the limit of more complex sequences.