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The n-th partial sum of the harmonic series, which is the sum of the reciprocals of the first n positive integers, diverges as n goes to infinity, albeit extremely slowly: The sum of the first 10 43 terms is less than 100 .
The nth partial sum of the series is the triangular number = = (+), which increases without bound as n goes to infinity. Because the sequence of partial sums fails to converge to a finite limit, the series does not have a sum.
Generalizing this argument, any infinite sum of values of a monotone decreasing positive function of (like the harmonic series) has partial sums that are within a bounded distance of the values of the corresponding integrals. Therefore, the sum converges if and only if the integral over the same range of the same function converges.
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (,,, …) defines a series S that is denoted = + + + = =. The n th partial sum S n is the sum of the first n terms of the sequence; that is,
The harmonic number H n can be interpreted as a Riemann sum of the integral: + = (+). The n th harmonic number is about as large as the natural logarithm of n . The reason is that the sum is approximated by the integral ∫ 1 n 1 x d x , {\displaystyle \int _{1}^{n}{\frac {1}{x}}\,dx,} whose value is ln n .
The geometric series is an infinite series derived from a special type of sequence called a geometric progression.This means that it is the sum of infinitely many terms of geometric progression: starting from the initial term , and the next one being the initial term multiplied by a constant number known as the common ratio .
By taking to be the partial sum function associated to some sequence, this leads to the summation by parts formula. Examples Harmonic numbers. If = for and ...
In mathematics, a telescoping series is a series whose general term is of the form = +, i.e. the difference of two consecutive terms of a sequence ().As a consequence the partial sums of the series only consists of two terms of () after cancellation.