Ali 2 said:I know about that .. the solution will be obtained easily by that method...but...could you solve the question without stirling's approximation ?
\frac{n}{(n!)^{\frac1n}}?The limit of a mathematical expression represents the value that the expression approaches as the variable (in this case, n) gets infinitely close to a certain value. In this case, finding the limit of \frac{n}{(n!)^{\frac1n}} can help us understand the behavior of the expression as n gets larger and larger.
\frac{n}{(n!)^{\frac1n}}?To find the limit, we can use a mathematical tool called L'Hôpital's rule which states that the limit of an indeterminate form (such as \frac{n}{(n!)^{\frac1n}}) can be found by taking the derivative of the numerator and denominator separately and then evaluating the limit again.
\frac1n exponent in the denominator?The \frac1n exponent in the denominator represents the root of the factorial function. This means that as n increases, the factorial function will grow at a slower rate and the expression will approach a finite limit.
\frac{n}{(n!)^{\frac1n}} be evaluated without using L'Hôpital's rule?Yes, the limit can also be evaluated using other mathematical techniques such as the squeeze theorem or the ratio test. However, L'Hôpital's rule is often the most efficient and straightforward method.
\frac{n}{(n!)^{\frac1n}} approaches?Yes, the limit of \frac{n}{(n!)^{\frac1n}} approaches 0 as n approaches infinity. This can be proven using L'Hôpital's rule and the fact that the factorial function grows much faster than the linear function n.