1 - 2 + 3 - 4 + 5 - 6 + ... made easy

Wolfram Alpha has made even simpler, and (more correct proof)<p><a href="http://www.wolframalpha.com/input/?i=1+-+2+%2B+3+-+4+%2B+5+-6+%2B+.." rel="nofollow">http://www.wolframalpha.com/input/?i=1+-+2+%2B+3+-+4+%2B+5+-...</a>.

I alway prefer this type of approach for solving problems. Even when formal proofs are required, I usually get the answer in this way and then reverse engineer the proof.<p>Another example (the sum of 1/2 + 1/4 + 1/8 + 1/16...): <a href="http://media-2.web.britannica.com/eb-media/79/26979-004-CF3F4DA2.gif" rel="nofollow">http://media-2.web.britannica.com/eb-media/79/26979-004-CF3F...</a>

I find it more helpful to note that<p><pre><code> 1/(1 + x)^2 = 1 - 2x + 3x^2 - 4x^3 + ... </code></pre> for |x| &#60; 1 and then to extend the summation to other values of x.

All such attempts for summation are ill-defined. The Abel summation is actually not a property - it is a <i>functional</i>. That said you take the function f(n)=&#60;n-th summand&#62;. Then you can define AS(f), but it shares very few properties with normal summation. If you are not careful you can easily reach contradiction.<p>For example<p>1/4 = <i>1 -2</i> +3 <i>-4 +5</i> -6 <i>+7</i> ... = //comment: 1-2-4+5=0,<p>3-6+9-12... =<p>3\sum i(-1)^i= 3* 1/4<p>Contradiction!<p>Edit: The original contradiction example was more complicated.

I personally found the part of the article about Abel summation[1] much more enlightening. It really seems to indicate <i>why</i> 1 - 2 + 3 ... = 1/4 instead of just showing that 1/4 arises if you look at the sums a certain way.<p>[1]: <a href="http://en.wikipedia.org/wiki/1_−_2_%2B_3_−_4_%2B_·_·_·#Abel_summation" rel="nofollow">http://en.wikipedia.org/wiki/1_−_2_%2B_3_−_4_%2B_·_·_·#Abel_...</a>

<a href="http://en.wikipedia.org/wiki/Telescoping_series" rel="nofollow">http://en.wikipedia.org/wiki/Telescoping_series</a>

<i>I don't know if a real mathematician would approve of my method, but at least it's easy to understand.</i><p>The good mathematician may not approve.<p>The better mathematician probably would approve.<p>The best mathematician would seek and codify a principal to describe the general case of the instance you cite.

It diverges under the ordinary definitions of infinite series, but one can generalize the notion of summation a little bit to allow for certain divergent series to have finite sums.