19th Century
The 19th century is remembered as the approximate period of Cauchy's and Abel's largely successful ban on the use of divergent series, but Grandi's series continued to make occasional appearances. Some mathematicians did not follow Abel's lead, mostly outside of France, and British mathematicians especially took "a long time" to understand the analysis coming from the continent.
In 1803, Robert Woodhouse proposed that 1 − 1 + 1 − 1 + · · · summed to something called
which could be distinguished from 1⁄2. Ivor Grattan-Guinness remarks on this proposal, "… R. Woodhouse … wrote with admirable honesty on the problems which he failed to understand. … Of course, there is no harm in defining new symbols such as 1⁄1+1; but the idea is 'formalist' in the unflattering sense, and it does not bear on the problem of the convergence of series."
Read more about this topic: History Of Grandi's Series