What is the Function for the Value of a Convergent Series Sum?

• I
• mathman
In summary, a convergent series sum is a mathematical concept where the sum of a sequence of numbers approaches a finite limit as the number of terms increases. It is calculated by taking the limit as the number of terms approaches infinity using methods such as the Ratio Test, Root Test, or Comparison Test. The main difference between a convergent and divergent series sum is that a convergent sum has a finite limit while a divergent sum does not. Some real-life applications of a convergent series sum include calculating energy in physics and returns on investments in finance. Infinite series sums also have practical uses in calculus, computer science, and statistics.
mathman
TL;DR Summary
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##?

Janosh89
No, there is not. A closed expression for ##\sum_{n=1}^\infty \frac{1}{n^3}## is not known. This is Apéry's constant. See https://en.wikipedia.org/wiki/Apéry's_constant for more information.

mfb said:
For some values there are analytic expressions. It's the Riemann zeta function.
I should have known! It is the zeta function for all ##c\gt 1##.

What is a convergent series sum?

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.

How is a convergent series sum calculated?

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.

What is the significance of a convergent series 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.

Can a series have more than one convergent sum?

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.

How is the convergence of a series determined?

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.

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