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eddybob123
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Hi all, is there any way to find what the largest value of n is such that $$\sum_{k=1}^\infty\frac{1}{k^n}$$ is divergent? I don't need an answer, I need an approach to the problem.
eddybob123 said:Hi all, is there any way to find what the largest value of n is such that $$\sum_{k=1}^\infty\frac{1}{k^n}$$ is divergent? I don't need an answer, I need an approach to the problem.
The Divergent Series is a mathematical concept that refers to an infinite series in which the terms grow larger and larger, rather than approaching a specific value. This means that the series does not have a finite sum and instead diverges.
To find the largest value of n in a Divergent Series, you can use the limit comparison test. This involves comparing the given series to a known series with known convergence or divergence properties. The largest value of n will be when the given series is equal to or greater than the known series.
Finding the largest value of n is important in a Divergent Series because it helps us understand the behavior of the series and determine whether it converges or diverges. This information can be useful in various mathematical and scientific applications.
Some common examples of Divergent Series include the Harmonic Series, the Geometric Series with a ratio greater than 1, and the Alternating Harmonic Series. These series have been extensively studied and their divergence has been proven using various mathematical techniques.
Yes, there are several real-world applications of Divergent Series in fields such as physics, engineering, and economics. For example, in physics, Divergent Series are used to model the behavior of infinite systems, such as the energy levels of atoms. In economics, Divergent Series can be used to model the growth and decay of populations or economies over time.