For geometric series there is the basic formula
For any series that can be transformed into this form we thus find its sum.
Example:
Some people prefer to remember a simpler formula for sum of geometric series,
without a and with
It is simple to get to this simplified version on the left from the general case, the constant a can be factored out and the beginning n0 can be fixed by factoring out the lowest power present in the sum (which is a trick specific for geometric series, see Important examples in Theory - Introduction) or by the general subtract-add trick, when we fill in the missing terms of the series and then subtract them.
The first method - the factoring out approach - is usually superior, which is not surprising (specific methods usually give better results than generic ones). Note that the factoring out trick is equivalent to reindexing the series, see the example in Summing up series in Theory - Introduction.
For other examples see this problem and this problem in Solved Problems - Summing up series.
Telescopic series
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series