Jacob_ said:Homework Statement
For which p > 0 does the sum
[itex]\displaystyle\sum\limits_{k=10}^∞ \frac{1}{k^p(ln(ln(k)))^p}[/itex]
converge?
Homework Equations
1/k^p converges for p > 1.
The Attempt at a Solution
I'm not really sure where to start. I want to use a comparison test with the p-series, but ln(ln(k)) < 1 for k < e^e, so the equation isn't greater or less than 1/k^p for the entire sum interval.
Convergence of a sum means that as we add more and more terms, the sum approaches a finite value. In other words, the sum does not grow infinitely large, but instead reaches a stable value.
The value of p in the question "For which values of p does this sum converge?" refers to the power of the term being added in the sum. This value is crucial in determining whether the sum will converge or not.
There are various mathematical tests and techniques that can be used to determine the convergence of a sum for a given value of p. Some common methods include the ratio test, comparison test, and integral test.
Yes, it is possible for a sum to converge for certain values of p and diverge for others. This depends on the specific series and the convergence tests used to analyze it.
Understanding the convergence of sums is crucial in many fields of science, such as physics, engineering, and economics. It helps in analyzing and predicting the behavior of systems that involve continuous change, such as motion, electricity, and financial markets.