most recent 30 from http://mathoverflow.net 2013-05-25T01:11:58Z http://mathoverflow.net/feeds/user/7972 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/30156/demystifying-complex-numbers/33632#33632 Alex Eremenko 2010-07-28T08:30:51Z 2010-07-28T08:30:51Z

<p>Consider the function f(x)=1/(1+x^2) on the real line. Using the geometric progression formula, you can expand f(x)=1-x^2+... . This series converges for |x|&lt;1 but diverges for all other x. Why this is so? The function looks nice and smooth everywhere on the real line.</p> <p>This example is taken from the Introduction of the textbook by B. V. Shabat.</p>