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Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms a n converge to 0 monotonically, but this condition is not necessary for convergence.
For instance, rearranging the terms of the alternating harmonic series so that each positive term of the original series is followed by two negative terms of the original series rather than just one yields [34] + + + = + + + = + + + = (+ + +), which is times the original series, so it would have a sum of half of the natural logarithm of 2. By ...
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation , named after Niels Henrik Abel who introduced it in 1826.
and the arithmetic means of these partial sums are: 1, 0, 2 ... , the alternating series of triangular numbers; its Abel and Euler sum is 1 ...
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
This means the partial sums of an alternating series also "alternates" above and below the final limit. More precisely, when there is an odd (even) number of terms, i.e. the last term is a plus (minus) term, then the partial sum is above (below) the final limit.
Adding or subtracting two series term-by-term, Multiplying through by a scalar term-by-term, "Shifting" the series with no change in the sum, and; Increasing the sum by adding a new term to the series' head. These are all legal manipulations for sums of convergent series, but 1 − 1 + 1 − 1 + · · · is not a convergent series.
Euler–Boole summation is a method for summing alternating series. The concept is named after Leonhard Euler and George Boole. Boole published this summation method, using Euler's polynomials, but the method itself was likely already known to Euler. [1] [2] Euler's polynomials are defined by [1]