The Fascination of Infinite Series

A series is obtained from a sequence by adding up its terms one by one. From a finite sequence a₁, a₂, a₃, …, a _n we obtain the finite series, or sum, a₁ + a₂ + a₃ + …, + a _n . But for a_ninfinite sequence a₁, a₂, a₃, …, a _n , …, a problem arises: How should we compute its sum? We cannot, of course, actually add up all its infinitely many terms; but we can, instead, sum up a finite, but ever increasing, number of terms: a₁, a₁ + a₂, a₁ + a₂ + a₃, and so on. In this way we obtain a new sequence, the sequence of partial sums of the original sequence. For example, from the sequence 1, 1/2, 1/3, …, 1/n, … we get the sequence of partial sums 1, 1 + 1/2 = 1.5, 1 + 1/2 + 1/3 = 1.83333 …, and so on. If this sequence of partial sums converges to a limit S, then we say that the infinite series a₁ + a₂ + a₃ + … converges to the sum S. For the sake of brevity, we also use the phrase, “the series has the (infinite) sum S.”