Convergence of series: quick revision, 19 bites

What Does It Mean for a Series to Converge?

The sum of infinitely many terms can be finite.

A **series** is the sum of the terms of a sequence: $\sum_{n=1}^{\infty} a_n$. It **converges** if the sequence of **partial sums** $S_N = \sum_{n=1}^{N} a_n$ approaches a finite limit as $N \to \infty$. If the limit does not exist (or is infinite), the series **diverges**.

Takeaway. Convergence means the partial sums approach a finite limit.

The Divergence Test: A Quick Check

If terms don't vanish, the series can't converge.

If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ **diverges**. This is the **Divergence Test** (also called the $n$th-term test). It is a necessary condition for convergence: if the terms do not approach zero, the series cannot converge.

Takeaway. Always check the limit of the terms first; if it's not zero, the series diverges.

Geometric Series: The Classic Example

The only series with a simple closed-form sum.

A **geometric series** converges if and only if $|r| < 1$. The sum is $\frac{a}{1-r}$. If $|r| \geq 1$, the series diverges (unless $a=0$, trivial).

Takeaway. Identify geometric series and use the ratio condition to determine convergence instantly.

Confusing Sequence vs. Series

A sequence is not a series.

A **sequence** is a list of numbers $a_1, a_2, a_3, \ldots$; a **series** is the sum of those numbers. Convergence of the sequence $a_n$ (i.e., $a_n \to 0$) does **not** imply convergence of the series $\sum a_n$.

Takeaway. Don't confuse the limit of terms with the sum of the series.

The Integral Test: Linking Series and Integrals

Use an improper integral to test convergence.

If $f(x)$ is positive, continuous, and decreasing for $x \geq 1$, and $a_n = f(n)$, then the series $\sum_{n=1}^{\infty} a_n$ and the improper integral $\int_{1}^{\infty} f(x) \, dx$ either both converge or both diverge.

Takeaway. The integral test converts a series into an integral, which is often easier to evaluate.

p-Series: A Family with a Simple Rule

Converges if p > 1, diverges otherwise.

A **p-series** is of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$. It converges if $p > 1$ and diverges if $p \leq 1$. This is a direct consequence of the integral test.

Takeaway. Memorize the p-series rule: it's a quick benchmark for many comparison tests.

Direct Comparison Test: Bounding Your Series

Compare with a known convergent or divergent series.

For series with nonnegative terms, if $0 \leq a_n \leq b_n$ for all $n$ (eventually), then: - If $\sum b_n$ converges, so does $\sum a_n$. - If $\sum a_n$ diverges, so does $\sum b_n$.

Takeaway. Compare to a known series to establish convergence or divergence.

Limit Comparison Test: When Direct Comparison Fails

Compare using limits instead of inequalities.

If $a_n, b_n > 0$ and $\lim_{n \to \infty} \frac{a_n}{b_n} = c$ where $0 < c < \infty$, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

Takeaway. Use limit comparison when direct inequalities are hard to establish.

Ratio Test: Best for Factorials and Exponentials

Look at the limit of the ratio of consecutive terms.

The **Ratio Test** is particularly effective when terms involve factorials, exponentials, or powers. It is inconclusive when the limit equals 1.

Takeaway. Reach for the ratio test when you see factorials or exponentials.

Root Test: Useful for nth Powers

Take the nth root of the absolute value.

The **Root Test** is handy when the entire term is raised to the $n$th power, e.g., $a_n = \left(\frac{n}{2n+1}\right)^n$.

Takeaway. Use the root test when terms are of the form $(b_n)^n$.

Absolute vs. Conditional Convergence

Absolute convergence is stronger than conditional.

A series $\sum a_n$ **converges absolutely** if $\sum |a_n|$ converges. It **converges conditionally** if $\sum a_n$ converges but $\sum |a_n|$ diverges. Absolute convergence implies convergence, but not vice versa.

Takeaway. Always check absolute convergence first; it's a stronger property.

Alternating Series Test: Special Case

For alternating terms, check decreasing to zero.

An **alternating series** $\sum (-1)^{n} b_n$ (with $b_n > 0$) converges if: 1. $b_n$ is decreasing (eventually), and 2. $\lim_{n \to \infty} b_n = 0$.

Takeaway. For alternating series, verify decreasing to zero to conclude convergence.

Why Convergence Matters in Real Life

Infinite sums appear in physics, finance, and engineering.

Series convergence is crucial in **Taylor series** for approximating functions, in **Fourier series** for signal processing, and in **compound interest** calculations. A divergent series would give meaningless infinite values, so knowing convergence ensures valid models.

Takeaway. Convergence guarantees that infinite sums yield finite, usable results.

Choosing the Right Test

A flowchart can guide your test selection.

Takeaway. Systematically narrow down the test based on the form of the terms.

Remembering the p-Series Rule

p > 1 converges, p ≤ 1 diverges.

Think: **p** for **power**. If the power is greater than 1, the terms shrink fast enough to sum to a finite value. If the power is 1 or less, the terms don't shrink fast enough.

Takeaway. p > 1 means the series converges; p ≤ 1 means it diverges.

The Harmonic Series Diverges Slowly

It grows like the natural logarithm.

The harmonic series $\sum \frac{1}{n}$ diverges, but its partial sums grow very slowly—like $\ln n$. It takes about $e^{100}$ terms to reach a sum of 100.

Takeaway. Divergence can be extremely slow, but it's still divergence.

Series as a Marathon

Each term is a step; the sum is the distance.

Think of a series as a marathon runner taking steps of decreasing length. If the steps shrink fast enough, the runner eventually stops at a finite distance (converges). If the steps shrink too slowly, the runner keeps going forever (diverges).

Takeaway. The rate at which terms shrink determines whether the total distance is finite.

Always State the Test Used

Show your reasoning clearly in exams.

When determining convergence, always explicitly state which test you are applying and verify its conditions. For example, for the ratio test, write: "Since $L = \lim \left| \frac{a_{n+1}}{a_n} \right| = 0.5 < 1$, the series converges absolutely."

Takeaway. Clear justification prevents errors and earns full credit.

Inconclusive Tests: Don't Force a Conclusion

Some tests fail; try another.

The ratio and root tests are inconclusive when $L = 1$. In such cases, do not conclude anything—use a different test, such as the comparison or integral test.

Takeaway. When a test is inconclusive, switch to another method.