Search results
Results from the WOW.Com Content Network
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.
In mathematical analysis, the alternating series test proves that an alternating series is convergent when its terms decrease monotonically in absolute value and approach zero in the limit. The test was devised by Gottfried Leibniz and is sometimes known as Leibniz's test , Leibniz's rule , or the Leibniz criterion .
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, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point.
Color representation of the Dirichlet eta function. It is generated as a Matplotlib plot using a version of the Domain coloring method. [1]In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: = = = + +.
If an equation can be put into the form f(x) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x 1 in the basin of attraction of x, and let x n+1 = f(x n) for n ≥ 1, and the sequence {x n} n ≥ 1 will converge to the solution x.
The Riemann zeta function ζ(s) is one of the most significant functions in mathematics because of its relationship to the distribution of the prime numbers. The zeta function is defined for any complex number s with real part greater than 1 by the following formula: ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle \zeta (s)=\sum _{n=1}^{\infty ...
An ODE problem can be expanded with the auxiliary variables which make the power series method trivial for an equivalent, larger system. Expanding the ODE problem with auxiliary variables produces the same coefficients (since the power series for a function is unique) at the cost of also calculating the coefficients of auxiliary equations.