Question

Q: Determine the following limits:
(a) lim((3√n)1/2n), (b) lim((n+1)1/ln(n+1)).

Answer 1

Answer on Question #64530 – Math – Real Analysis

Question

Determine the following limits:

(a) $\lim((3\nu n)1/2n)$, (b) $\lim((n+1)1/ln(n+1))$

Solution

(a) $\lim_{n\to \infty}\frac{3\sqrt{n}}{2n} = \frac{3}{2}\lim_{n\to \infty}\frac{1}{\sqrt{n}} = 0$

(b) We use the Stolz-Cesàro theorem:

$$\lim _ {n \to \infty} \frac {a _ {n}}{b _ {n}} = \lim _ {n \to \infty} \frac {a _ {n} - a _ {n - 1}}{b _ {n} - b _ {n - 1}},$$

where $a_{n} = n + 1$, the sequence $b_{n} = \ln (n + 1)$ is strictly monotone (strictly increasing) and divergent (approaches $+\infty$).

So we have

$$\lim _ {n \to \infty} \frac {n + 1}{\ln (n + 1)} = \lim _ {n \to \infty} \frac {(n + 1) - n}{\ln (n + 1) - \ln n} = \lim _ {n \to \infty} \frac {1}{\ln \left(\frac {n + 1}{n}\right)} = \frac {1}{\ln \left(\lim _ {n \to \infty} \left(1 + \frac {1}{n}\right)\right)} = 1 / \ln 1 = \infty .$$

Answer: (a) 0; (b) $\infty$.

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