A couple of notes on convergent series

Note: I originally published this entry on a Google Sites page in September 2013, but I have moved it hear as I hope to make better use of the $latex \LaTeX$ support offered by wordpress.com

 

I completed the first year of my maths degree in June 2013. I studied Analysis in all three terms (sequences and series in the first term, continuity and differentiability in the second, and Riemann integration in the third) and feel I have learnt a lot. Occasionally, amongst all this theory, I must admit to sometimes having lost touch with the original questions that provoked its development.

Consider the following naïve approach to infinite series. What is the answer if I add 1 and -1 alternately forever? That is

$latex \displaystyle{1+(-1)+1+(-1)+ … = ???}$

If I bracket the terms in pairs starting with the first and second terms, the answer appears to be 0:
$latex…