10.1 Defining Convergent and Divergent Infinite Series
Keywords
Infinite Series 无穷级数 • Partial Sum 部分和
Convergence 收敛 • Divergence 发散 • Limit of Sequence 数列极限
From Sequences to Series
A sequence is an ordered list of numbers \(a_1, a_2, a_3, \dots, a_n\). An infinite series is the sum of those numbers:
But how do we add up an infinite number of things? We do it by looking at the limit of "partial chunks" of the sum.
The Partial Sum (\(S_n\))
Core Concept: The \(n\)-th Partial Sum (LIM-7.A.1)
The \(n\)-th partial sum, denoted by \(S_n\), is the sum of the first \(n\) terms of the series:
As we add more terms, we create a new sequence called the sequence of partial sums: \(\{S_1, S_2, S_3, \dots, S_n, \dots\}\).
Defining Convergence
Core Concept: Convergence vs. Divergence (LIM-7.A.2)
An infinite series \(\sum\limits_{n=1}^{\infty} a_n\) converges to a real number \(S\) if and only if the limit of its sequence of partial sums exists and equals \(S\):
If the limit \(\lim\limits_{n \to \infty} S_n\) does not exist (e.g., it goes to \(\infty\) or oscillates), the series is said to diverge.
Visualizing Convergence
Consider the series \(\sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^n = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots\) The partial sums are:
- \(S_1 = 0.5\)
- \(S_2 = 0.75\)
- \(S_3 = 0.875\)
- \(S_4 = 0.9375\)
As \(n \to \infty\), \(S_n \to 1\).
Extension: The Riemann Zeta Function
Series are not just for calculus homework; they are at the heart of number theory. The Riemann Zeta Function, \(\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}\), involves summing infinite terms.
The famous Riemann Hypothesis (one of the Millenium Prize Problems) concerns the zeros of this function. Proving it would unlock deep secrets about the distribution of prime numbers! For now, just know that when \(s=1\), the series \(\sum \frac{1}{n}\) is called the Harmonic Series, and it actually diverges to \(\infty\), even though the terms being added get smaller and smaller.