We don't consider the infinite series which doesn't converge. We consider a limited number of the first terms in order to derive an equivalent. This is the usal way for asymptotic series.
The sum 1/log(n) is a mathematical series that represents the harmonic series with natural logarithmic terms. It is commonly used in mathematical analysis and has applications in fields such as number theory and physics.
When we say that \sum 1/log(n) goes like something, we mean that it has a similar behavior or trend as that particular value or function. In other words, as n increases, the sum 1/log(n) will approach the same limit or follow a similar pattern as the specified value or function.
The asymptotic behavior of \sum 1/log(n) is logarithmic, meaning that as n increases, the sum will approach infinity at a slower rate than other types of functions such as polynomial or exponential functions.
The sum 1/log(n) is closely related to the natural logarithm function, ln(n). In fact, as n approaches infinity, the sum 1/log(n) approaches the same limit as ln(n), which is infinity. This means that the growth rate of both functions is equivalent.
No, the sum 1/log(n) will always approach infinity as n increases. This is because the terms in the series decrease very slowly, and as n increases, there are an infinite number of terms in the series, making the sum infinite. However, for certain values of n, the sum may appear to approach a finite value, but it will never actually reach it.
