The alternating series is Σ n = 1 ∞ j n f ( n ) = [ j ] f \Sigma _{n = 1}^\infty {j_n}f(n) = [j]f , with f a single-signed monotonic function of the real variable x. The j n {j_n} are ± 1 \pm 1 , their sign fixed by repetition of the ’template’ [j] of finite length 2p. [j] constitutes a difference scheme of ’differential order’ D, which can be determined. The principal theorem is that [ j ] f [j]f is ’partially convergent’ if and only if lim x → ∞ f ( D − 1 ) ( x ) {\lim _{x \to \infty }}{f^{(D - 1)}}(x) is bounded. A series is partially convergent when the limit as M → ∞ M \to \infty of the sum of 2pM terms exists. For [j] ’pure’, the improved Euler-Maclaurin expansion (IEM) gives the compact representation (A) S ( p ) ∼ − μ ( D ) 2 p ∑ r = 0 ∞ ( 2 p ) 2 r B 2 r ( 1 / 2 ) ( 2 r ! ) f ( 2 r + D − 1 ) ( θ r ) , 1 − p ⩽ θ r ⩽ p . \begin{equation}\tag {A} S^{(p)} \sim - \frac {\mu (D)}{2p} \sum _{r = 0}^\infty (2p)^{2r} \frac {B_{2r}(1/2)}{(2r!)} f^{(2r+D-1)} (\theta _r),\quad 1 - p \leqslant \theta _r \leqslant p. \end{equation} S ( p ) S^{(p)} is the sum, μ ( D ) \mu (D) is the Dth ’template moment’, and the B 2 r {B_{2r}} are Bernoulli numbers. Efficient means for practical summation of these series follow also from IEM. In illustration, 10 alternating series with D ranging from 1 to 3 are summed using IEM. It is found that the leading term of (A) with θ 0 = 1 / 2 \theta _0 = 1/2 gives a simple but effective estimate of sums. The paper also gives a comparison with Euler’s transformation in the case p = 1 p = 1 and discusses sums to N terms with N / 2 p N/2p nonintegral and finite but large.