1. Number TheoryIn mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a1,a2,a3,…) defines a series S that is denoted S=a1+a2+a3+⋯=∑k=1∞ak. The nth partial sum Sn is the sum of the first n terms of the sequence; that is, Sn=a1+a2+⋯+an=∑k=1nak. A series is convergent (or converges) if the sequence (S1,S2,S3,…) of its partial sums tends to a limit and that limit is finite; that means that, when adding one ak after the other in the order given by the indices, the partial sums approximate a number ℓ. More precisely, for any given error tolerance, all but finitely many of the partial sums lie within that tolerance of the value ℓ. If the series is convergent, the (necessarily unique) number ℓ is called the sum of the series. The same notation ∑k=1∞ak is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: a + b denotes the operation of adding a and b as well as the result... from wikipedia.org