The infinite series whose terms are the natural numbers 1 + 2 + 3 + 4 + ⋯ is a divergent series. The nth partial sum of the series is the triangular number ∑ k = 1 n k = n ( n + 1 ) 2 , {\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}},} which increases without...

In mathematics, the infinite series 1 − 1 + 1 − 1 + ⋯, also written ∑ n = 0 ∞ ( − 1 ) n {\displaystyle \sum _{n=0}^{\infty }(-1)^{n}} is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent series, meaning that it lacks a sum in the usual sense...

== Geometry and infinite zeros == === Grandi === Guido Grandi (1671–1742) reportedly provided a simplistic account of the series in 1703. He noticed that inserting parentheses into 1 − 1 + 1 − 1 + · · · produced varying results: either ( 1 − 1 ) + ( 1 − 1 ) + ⋯ = 0 {\displaystyle (1-1)+(1-1)+\cdots =0} or 1 + ( − 1 + 1 ) + ( − 1...

Compartmental models are a very general modelling technique. They are often applied to the mathematical modelling of infectious diseases. The population is assigned to compartments with labels – for example, S, I, or R, (Susceptible, Infectious, or Recovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again. The origin of such models is the early 20th century, with important works being that of Ross in 1916, Ross and Hudson in 1917, Kermack...