What is the Limit of the Partial Sum of a Series?

The limit of the partial sum of a series, if it exists, is the sum of the series.

Here's a breakdown:

Understanding Series and Partial Sums

A series is the sum of the terms of a sequence. For example, if we have the sequence {a₁, a₂, a₃,...}, the corresponding series is a₁ + a₂ + a₃ + ... .

The n-th partial sum (S_n) of a series is the sum of its first n terms:

S_n = a₁ + a₂ + a₃ + ... + a_n

So, we have a sequence of partial sums: {S₁, S₂, S₃,...}.

Convergence and Divergence

The key question is: What happens to the sequence of partial sums as n approaches infinity?

• Convergence: If the sequence of partial sums {S_n} approaches a finite limit L as n goes to infinity, we say the series converges, and its sum is L. Mathematically:

  lim_n→∞ S_n = L
  In this case, L is the "sum" of the infinite series.

• Divergence: If the sequence of partial sums does not approach a finite limit (it either oscillates or grows without bound), we say the series diverges. In this case, the series has no sum in the traditional sense.

Examples

• Convergent Series: Consider the geometric series 1/2 + 1/4 + 1/8 + 1/16 + ... The sequence of partial sums is {1/2, 3/4, 7/8, 15/16,...}. This sequence converges to 1. Therefore, the sum of this infinite series is 1.

• Divergent Series: Consider the series 1 + 1 + 1 + 1 + ... The sequence of partial sums is {1, 2, 3, 4,...}. This sequence grows without bound (approaches infinity). Therefore, the series diverges and does not have a finite sum.

In Summary

The limit of the partial sum of a series is the value the sum "approaches" as you add more and more terms. If this limit exists and is a finite number, that number is defined as the sum of the infinite series. If no such limit exists, the series diverges and does not have a sum.