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Summing up series of real numbers is a difficult task. There are essentially only two kinds of series that can be summed up without specific tricks or advanced methods:
One advanced trick that can be helpful applies to
Often we encounter series that we cannot sum up, then we would like to approximate the sum. In such situations it is useful to be able to estimate the error of approximation.
When manipulating series we use various tricks. The most useful ones are covered in Manipulating series in Methods Survey - Series of functions.