Difference Between Convergence and Divergence

Partial Sums Determine Whether a Series Converges

Adding the terms of an infinite sequence produces an infinite series. We determine its behavior through partial sums: if they approach a finite number, the series is convergent. Otherwise it is divergent.

Convergence of Sequences

A sequence $(a_n)$ converges to $L$ when its terms can be made arbitrarily close to $L$ by taking $n$ sufficiently large. We write:

If no finite limit exists, the sequence diverges. Divergence can take different forms:

• $a_n=n$ grows without bound.
• $a_n=-n$ decreases without bound.
• $a_n=(-1)^n$ oscillates between two values.

From a Sequence to a Series

For an infinite series:

To determine convergence, we study the sequence of partial sums:

The series converges exactly when the sequence of partial sums converges. The limit of the terms $a_n$ alone does not decide whether the series converges.

Convergent Series

A series is convergent when its partial sums approach a finite value. In an idealized bouncing-ball model, for example, successively shorter paths can have a finite total length.

Partial Sums Settle on One Limit

A convergent series has two properties that follow from its partial sums. The partial sums approach one real number, and the terms must shrink toward zero.

• 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 $n$th term ($u_n$) must approach $0$ as $n$ approaches infinity ($\lim_{n \to \infty} u_n = 0$).

Examples of Convergent Series

A geometric series converges whenever the absolute value of its ratio is smaller than $1$. Take the series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$, whose ratio is $\frac{1}{2}$. Its partial sums keep closing in on $2$, and the limit formula confirms that value directly.

A ratio of $1$ or larger leaves the series without a finite sum.

Divergent Series

A series is divergent when its partial sums do not approach a finite value. The partial sums may:

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

Partial Sums Fail to Settle

A divergent series either has no single limit or moves without bound. The two conditions below separate those cases.

• Its partial sum ($S_n$) does not approach a specific value $L$ as $n$ approaches infinity.
• If $u_n\not\to 0$, then the series is definitely divergent. This covers both a nonzero term limit and a term sequence that has no limit.

Series That Diverge

The three families below diverge for different reasons. The harmonic series shows that shrinking terms alone cannot force convergence.

• Arithmetic Series (except $0 + 0 + \dots$): Their partial sums grow toward $\infty$ or decrease toward $-\infty$.

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

• Geometric Series with $|r| \ge 1$:

  • If $r \ge 1$, its partial sums approach $\pm \infty$, depending on the sign of the first term.

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

  • If $r \le -1$, its partial sums oscillate.

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

• Harmonic Series: In $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots$, the $n$th term $u_n = \frac{1}{n}$ approaches $0$, but the partial sums still grow toward infinity $\infty$. Thus, $u_n \to 0$ is necessary but not sufficient for convergence.

Complete Classification of Geometric Series

Consider the series:

For $a\ne0$, every possible ratio falls into one of these cases:

• If $|r|<1$, the series converges to $a/(1-r)$.
• If $r=1$, the partial sums are $na$ and diverge.
• If $r=-1$, the partial sums alternate between two values.
• If $|r|>1$, even the terms fail to approach zero.

A divergent series therefore does not always have the value $\infty$. Oscillation and unbounded negative growth are different forms of divergence.

Sequences That Move One Way and Stay Limited

Every monotone and bounded sequence converges. This criterion is useful when a limit is difficult to read directly from a formula.

The partial sums of a series with non-negative terms are increasing. If those partial sums are also bounded above, the series converges.

Comparing Convergent and Divergent Series

The table collects the three tests, so the partial-sum behavior, the term condition, and a matching example can be read together.

Exercise

Decide whether each series converges or diverges. Check the term behaviour first, because a term that does not approach zero already settles the question. Write your conclusion for each series before you read the solution.

1. $\sum_{k=0}^{\infty}3(1/4)^k$
2. $\sum_{k=1}^{\infty}1$
3. $\sum_{k=0}^{\infty}(-1)^k$

Solution

1. The first series is geometric with $|r|=1/4<1$, so it converges:

   $$\frac{3}{1-1/4}=4$$

2. The terms of the second series do not approach zero, so the series diverges.

3. The partial sums of the third series alternate between two values. This oscillation prevents them from approaching one limit.