Nikitin said:Homework Statement
Does this converge or diverge?
Ʃ1/(1+2+3+4+5...+n), as n---> infinity?
The Attempt at a Solution
I rewrote this into Ʃ(Ʃ1/n) (is it correct?).
I figured that since Ʃ(1/n) diverges, then the sum of each partial sum most (obviously) also diverge.
However, it appears I'm mistaken. Can somebody help?
The harmonic series is an infinite series of numbers that consists of the reciprocals of positive integers. It can be represented by the formula 1 + 1/2 + 1/3 + 1/4 + ...
The sum of the sum of harmonic series is a divergent series, meaning that its sum is infinite. This can be proven through mathematical analysis and is a well-known concept in mathematics.
The sum of the sum of harmonic series can be calculated using a mathematical technique called the Euler-Maclaurin summation formula. This formula provides an approximation of the sum and can be used to show that the sum is infinite.
The sum of the sum of harmonic series has several applications in mathematics, including in the study of number theory and calculus. It also serves as a fundamental example of a divergent series and helps to illustrate the concept of infinity in mathematics.
While the sum of the sum of harmonic series may not have a direct real-world application, the concept of divergent series has been applied in various fields, such as physics and economics. Additionally, understanding the properties of divergent series can help in the development of mathematical theories and techniques.