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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.
Convergent series. 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. {\displaystyle S=a_ {1}+a_ {2}+a_ {3}+\cdots =\sum _ {k=1}^ {\infty }a_ {k}.} The n th partial sum Sn is the sum of the first n terms of the sequence; that is,
Calculus. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: The first terms of the series sum to approximately , where is the natural logarithm and is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it ...
Then the sum of the resulting series, i.e., the limit of the sequence of partial sums of the resulting series, satisfies +, = (, +,) =, +,, when the limits exist. Therefore, first, the series resulting from addition is summable if the series added were summable, and, second, the sum of the resulting series is the addition of the sums of the ...
Fejér's theorem. In mathematics, Fejér's theorem, [1][2] named after Hungarian mathematician Lipót Fejér, states the following: [3] Fejér's Theorem — Let be a continuous function with period , let be the nth partial sum of the Fourier series of , and let be the sequence of Cesàro means of the sequence , that is the sequence of ...
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 .
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. [1][2] The cancellation technique, with part of each term cancelling with part of the next term, is ...
The first four partial sums of 1 + 2 + 4 + 8 + ⋯. In mathematics, 1 + 2 + 4 + 8 + ⋯ is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term, 1, and its common ratio, 2. As a series of real numbers it diverges to infinity, so the sum of this series is infinity.