## MinutePhysics Maths Trolling (Adding Past Infinity)

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by Jay, 9 years ago

A response to the video "Adding past infinity" by MinutePhysics http://www.youtube.com/watch?v=kIq5CZlg8Rg
Unlike convergent series, divergent series do not have limits (the sum of the first n terms, are not getting closer to a particular value as n increases).
However, if we assume a divergent series does have a limit, then methods used to find limits of convergent series may also be applied to divergent series. Any value found in this manner can still be meaningful, and may be regarded as the sum of the divergent series.
Divergent series: http://en.wikipedia.org/wiki/Divergent_series
One such method (mentioned in the video) is to use the sequence of average partial sums (sum the series to the nth term then divide by n, and do this for all n). If the sum of these partial sums tends to a limit (i.e. form a convergent series) it is called Cesàro summable, see http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation
For geometric series, Cesàro summation is a fairly weak method. It works when r=-1, but it doesn't work on more difficult geometric sums, like when r=2 (as in this video).
However, other methods of summation of geometric series may be used. Any method that is regular, linear and stable (see http://en.wikipedia.org/wiki/Divergent_series#Properties_of_summation_methods) result in the same answer, namely a/(1-r).
More on divergent geometric series: http://en.wikipedia.org/wiki/Divergent_geometric_series
The exception is r=1. Yet, this too may be summed using methods that sacrifice of some nice properties (stability). See http://en.wikipedia.org/wiki/1_%2B_1_%2B_1_%2B_1_%2B_%C2%B7_%C2%B7_%C2%B7
This idea of extending the definition of sum to divergent series is not just for divergent geometric series. For example, the series 1+2+3+4+... can be calculated to be -1/12 http://en.wikipedia.org/wiki/1%2B2%2B3%2B4%2B
In general, such extensions are called Analytic Continuation http://en.wikipedia.org/wiki/Analytic_continuation