Understanding Faulhaber's Formula for Sum of Powers of n Integers

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SUMMARY

Faulhaber's Formula provides a method for calculating the sum of the pth powers of the first n integers. The formula can be effectively applied using the Euler-MacLaurin summation technique, which involves rewriting the sum as a Stieltjes integral and integrating by parts until the remaining integrand is zero. Understanding Bernoulli numbers is essential for applying Faulhaber's Formula correctly. Numerous references are available for both the Euler-MacLaurin summation and Bernoulli numbers to facilitate deeper comprehension.

PREREQUISITES
  • Understanding of Faulhaber's Formula
  • Familiarity with Euler-MacLaurin summation
  • Knowledge of Stieltjes integrals
  • Comprehension of Bernoulli numbers
NEXT STEPS
  • Research the derivation of Faulhaber's Formula
  • Study the Euler-MacLaurin summation technique in detail
  • Explore the properties and applications of Bernoulli numbers
  • Practice solving problems involving sums of powers using Faulhaber's Formula
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Mathematicians, educators, and students interested in advanced calculus and number theory, particularly those focusing on summation techniques and polynomial expressions.

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hi everybody, I have a question in math to figure out the general term for the sum of the pth powers of n integers. I found a formula called faulhabers formula to do this question, but I do not understand the method behind it. can someone please help me?
 
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You can prove it using Euler-MacLaurin summation, where you essentially rewrite the sum as a Stieltjes integral then integrate by parts enough time that the remaining integrand is zero (the p+1 derivitative of x^p vanishes). Look up Euler-MacLaurin summation (lot's of references).

Or are you trying to understand how to apply Faulhaber's in a given situation? It's just plug and chug, but you'll need to know what the Bernoulli numbers are (again, lots of references)
 

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