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- sum ##\frac{1}{n^c}## where ##c\gt 1##
##\sum_n \frac{1}{n^c}## converges for ##c\gt 1##. Is there an expression for the value of the sum as a function of ##c##?
I should have known! It is the zeta function for all ##c\gt 1##.mfb said:For some values there are analytic expressions. It's the Riemann zeta function.
A convergent series sum is a mathematical concept that refers to the sum of an infinite sequence of numbers that approaches a finite value as the number of terms in the sequence increases.
A convergent series sum can be calculated using various mathematical methods, such as the geometric series formula or the telescoping series method. It is important to note that not all series are convergent, and some may require more complex methods to determine their sum.
The convergence of a series allows us to make predictions and draw conclusions about the behavior of the series as a whole. It also has practical applications in fields such as physics, engineering, and economics, where infinite series are used to model real-world phenomena.
No, a series can only have one convergent sum. This is because the convergence of a series is determined by the behavior of the series as a whole, not just individual terms. If a series has more than one convergent sum, it would violate the fundamental principles of mathematics.
The convergence of a series is determined by analyzing the behavior of the terms in the series as the number of terms approaches infinity. If the terms approach a finite value, the series is said to be convergent. If the terms approach infinity, the series is said to be divergent.