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hivesaeed4
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Has anybody got any idea as to how to prove that Ʃ 1/(n(log(n))^p) converges? (where p>1)
hivesaeed4 said:Has anybody got any idea as to how to prove that Ʃ 1/(n(log(n))^p) converges? (where p>1)
DonAntonio said:First way, the integral test: [itex]\int_2^\inf \frac{1}{x\log^px} dx=\frac{\log^{1-p}(x)}{1-p}|_2^\inf \rightarrow \frac{log^{1-p}(2)}{p-1}[/itex] .
Second way, Cauchy's Condensation test: taking [itex]n=2^k[/itex] , the series's general term is [itex]\frac{1}{2^kk^p\log^p(2)}[/itex] , so multiplying this by [itex]2^k[/itex] we get [itex]\frac{1}{k^p\log^p(2)}[/itex] , which is a multiple of the series of [itex]\frac{1}{k^p}[/itex] , which we know converges for [itex]p>1[/itex] .
DonAntonio
Convergence in mathematics refers to the behavior of a sequence or series as its terms approach a specific limit. In other words, it is the process of approaching a fixed value or point as the number of iterations increases.
Proving convergence of 1/log(n) is important in mathematics as it helps in understanding the behavior of the logarithmic function and its relationship to other mathematical concepts. It also allows for the evaluation of the limit of 1/log(n) as n approaches infinity, which can have applications in various areas of mathematics and science.
To prove convergence of 1/log(n), one can use the Limit Comparison Test or the Integral Test. These tests compare the given series to a known series with a known convergence behavior, allowing for the determination of the convergence or divergence of the series in question.
The intuition behind the convergence of 1/log(n) lies in the fact that the logarithmic function grows very slowly, and as n approaches infinity, the value of 1/log(n) approaches 0. This can also be visualized by plotting the function, where it can be seen that as n increases, the curve approaches the x-axis.
Yes, there are several real-world applications of proving the convergence of 1/log(n), such as in the analysis of algorithms, population growth models, and in the study of computational complexity. It can also be used in various scientific fields, such as physics, to understand the behavior of certain phenomena as they approach a limit.