Difference Between Convergence and Divergence

What Are Convergent and Divergent Series?

In mathematics, when we sum the terms of an infinite sequence, we get an infinite series. The important question is: does this infinite sum approach a specific number (convergent) or not (divergent)?

Convergent Series

A series is called convergent if the sum of its terms approaches a finite value. Imagine a bouncing ball - the total distance it travels stops at one number, not continuing to increase without bound.

Characteristics of Convergent Series

• Its partial sum (the sum of the first $n$ terms, $S_n$) approaches a value $L$ as $n$ approaches infinity ($\lim_{n \to \infty} S_n = L$, where $L$ is a real number).
• Necessary condition (but not sufficient): the nth term ($u_n$) must approach 0 as $n$ approaches infinity ($\lim_{n \to \infty} u_n = 0$).

Examples of Convergent Series

• Geometric Series with $|r| < 1$: This is the most common example.

  For instance: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$. Its sum approaches 2.

  $$S_\infty = \frac{a}{1 - r} = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2$$

Divergent Series

A series is called divergent if the sum of its terms does not approach a finite value. Its sum could:

• Keep growing toward positive infinity ($\infty$).
• Keep decreasing toward negative infinity ($-\infty$).
• Oscillate between several values without ever settling.

Characteristics of Divergent Series

• Its partial sum ($S_n$) does not approach a specific value $L$ as $n$ approaches infinity.
• If $\lim_{n \to \infty} u_n \neq 0$ (the nth term does not approach 0), then the series is definitely divergent.

Examples of Divergent Series

• Arithmetic Series (except $0 + 0 + \dots$): Their sum always approaches $\infty$ or $-\infty$.

  For instance: $1 + 2 + 3 + 4 + \dots$ (approaches $\infty$)

• Geometric Series with $|r| \ge 1$:

  • If $r \ge 1$, its sum approaches $\pm \infty$ (depending on the sign of the first term).

    Example: $2 + 4 + 8 + 16 + \dots$ (approaches $\infty$)

  • If $r \le -1$, its sum oscillates.

    Example: $1 - 2 + 4 - 8 + \dots$ (Partial sums: $1, -1, 3, -5, \dots$ do not approach one value)

• Harmonic Series: $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots$. This is an interesting example. Although its nth term ($u_n = \frac{1}{n}$) approaches 0, the sum of the series still approaches infinity ($\infty$). This shows that the condition $u_n \to 0$ alone is not sufficient to guarantee convergence.

Summary of Key Differences