Sum of the powers of natural numbers

In summary, the "Sum of the powers of natural numbers" refers to the concept of adding together a series of numbers raised to different powers. The formula for calculating this sum is ∑n^k = (n+1)^(k+1) / (k+1), and it has various applications in mathematics. It differs from the "Sum of the first n natural numbers," which involves adding consecutive natural numbers, and it cannot be calculated for negative numbers.
  • #1
pyfgcr
22
0
Hi everyone. I have learned that:
1+2+3+...=[itex]\frac{n(n+1)}{2}[/itex]
12+22+32=[itex]\frac{n(n+1)(2n+1)}{6}[/itex]
I want to know what the general formula of Ʃna, in which n and a are natural numbers, respect to n and a.
 
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  • #2
See Faulhaber's formula (and the page about Bernoulli numbers, as they appear in the general formula).
 
  • #3
Actually, a simpler generalization of
1+2+3+ ... = n(n+1)/2
is
1.2 + 2.3 + 3.4 + ... = n(n+1)(n+2)/3
1.2.3 + 2.3.4 + 3.4.5 + ... = n(n+1)(n+2)(n+3)/4
etc.
 
  • #4
Now I know. Thanks for the answer.
 
  • #5


Hello, thank you for sharing your findings on the sums of powers of natural numbers. It is fascinating to see how these sums can be represented by simple formulas. To answer your question, the general formula for Ʃna would be:

Ʃna = \frac{n(n+1)^a}{a+1}

This formula can be derived using the method of finite differences, which involves finding patterns in the differences between consecutive terms in a series. In this case, the differences between consecutive terms in the series 1^a, 2^a, 3^a, ... are a, 2a, 3a, ... which can be represented as a linear series. Using the formula for the sum of a linear series, we can arrive at the general formula for Ʃna.

I hope this helps answer your question. Keep exploring and discovering new patterns in mathematics!
 

What is the "Sum of the powers of natural numbers"?

The sum of the powers of natural numbers refers to the mathematical concept of adding together the values of a series of numbers raised to different powers, starting from 1 and increasing by 1 for each subsequent term.

What is the formula for calculating the sum of the powers of natural numbers?

The formula for calculating the sum of the powers of natural numbers is ∑n^k = (n+1)^(k+1) / (k+1), where n is the last term in the series and k is the power being raised to.

What is the significance of the "Sum of the powers of natural numbers" in mathematics?

The sum of the powers of natural numbers has various applications in mathematics, including in number theory, calculus, and probability. It is also used in the study of polynomial equations and in the derivation of mathematical series.

What is the difference between the "Sum of the powers of natural numbers" and the "Sum of the first n natural numbers"?

The "Sum of the powers of natural numbers" involves adding terms that are raised to different powers, while the "Sum of the first n natural numbers" simply involves adding consecutive natural numbers. Additionally, the "Sum of the powers of natural numbers" has a formula for any value of n, while the "Sum of the first n natural numbers" has a specific formula for n.

Can the "Sum of the powers of natural numbers" be calculated for negative numbers?

No, the "Sum of the powers of natural numbers" is only applicable for positive natural numbers. For negative numbers or non-natural numbers, other mathematical concepts such as the geometric series or the Riemann zeta function can be used to calculate the sum of powers.

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