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anemone
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Find the
$\displaystyle\lim_{n\to\infty}\frac{2ln(2)+3ln(3)+...+nln(n)}{n^2ln(n)}$
$\displaystyle\lim_{n\to\infty}\frac{2ln(2)+3ln(3)+...+nln(n)}{n^2ln(n)}$
anemone said:Find the
$\displaystyle\lim_{n\to\infty}\frac{2ln(2)+3ln(3)+...+nln(n)}{n^2ln(n)}$
anemone said:Find the
$\displaystyle\lim_{n\to\infty}\frac{2ln(2)+3ln(3)+...+nln(n)}{n^2ln(n)}$
anemone said:Find the
$\displaystyle\lim_{n\to\infty}\frac{2ln(2)+3ln(3)+...+nln(n)}{n^2ln(n)}$
sbhatnagar said:$$\begin{align*}\lim_{n \to \infty}\sum_{r=1}^{n}\frac{r \ln(r)}{n^2 \ln(n)} &= \lim_{n \to \infty} \frac{1}{\ln(n)} \left[ \sum_{r=1}^{n}\frac{r}{n^2} \left\{ \ln{\left( \frac{r}{n}\right)+\ln(n)}\right\}\right] \\ &= \lim_{n \to \infty} \frac{1}{\ln(n)} \left[ \sum_{r=1}^{n}\frac{r}{n^2}\ln{\left( \frac{r}{n}\right)}+ \sum_{r=1}^{n}\frac{r\ln(n)}{n^2}\right] \\ &=
\lim_{n \to \infty}\frac{1}{\ln(n)} \left[ \int_{0}^{1}x \cdot \ln(x) \, dx + \frac{\ln(n)}{n^2}\sum_{r=1}^{n}r\right] \\
&=\lim_{n \to \infty}\frac{1}{\ln(n)} \left[ -\frac{1}{4} + \frac{n(n+1)\ln(n)}{2n^2}\right] \\
&=\lim_{n \to \infty} \left[ -\frac{1}{4\ln(n)}+\frac{1}{2}+\frac{1}{2n}\right] \\
&=0+\frac{1}{2}+0 \\
&=\frac{1}{2}
\end{align*}$$
CaptainBlack said:The replacement of the Riemann sum by the integral inside the limit when going from line 2 to 3 needs more justification.
CB
chisigma said:In $(0,1]$ the function $f(x)= x\ \ln x$ is bounded [more precisely $0 \le |f(x)| \le \frac{1}{e}$...] and continuous so that is Riemann-integrable...
Kind regards
$\chi$ $\sigma$
The natural logarithm, denoted as ln(x), is a mathematical function that is the inverse of the exponential function. It is used to find the exponent to which a base number must be raised to obtain a given number. In the context of limits, the natural logarithm is often used to evaluate the behavior of a function as it approaches a specific value.
To find the limit of a natural logarithm, we need to substitute the value that the function is approaching into the logarithm, simplify the expression, and then take the limit as the input value approaches the specified value. In some cases, we may need to use algebraic manipulations or L'Hopital's rule to evaluate the limit.
Some of the common properties of natural logarithms that are useful in finding limits include the fact that ln(1) = 0, ln(e) = 1, and ln(ab) = ln(a) + ln(b). Additionally, the natural logarithm function is continuous and differentiable for all positive real numbers.
Yes, there are some special cases when evaluating the limit of a natural logarithm. For example, if the input value is approaching 0, we will end up with an indeterminate form of ln(0), which cannot be evaluated. In such cases, we need to use techniques such as L'Hopital's rule or algebraic manipulations to evaluate the limit.
The limit of a natural logarithm can be used in various real-world applications, such as in finance, biology, and physics. In finance, it can be used to calculate compound interest and continuously compounded interest rates. In biology, it can be used to model population growth and decay. In physics, it can be used to describe the rate of change of a quantity over time.