Infinite Series

The central idea

An infinite series $\sum_{n=1}^{\infty}a_n$ is defined through its partial sums $S_N=\sum_{n=1}^Na_n$. The series converges to $S$ when $S_N\to S$; otherwise it diverges. The terms $a_n$ and partial sums $S_N$ are different sequences and must not be confused.

Definitions, hypotheses, and notation

Cauchy's criterion states that convergence occurs exactly when every sufficiently far tail $a_{m+1}+\cdots+a_n$ is arbitrarily small. This formulation explains why a finite prefix is irrelevant and why tail estimates certify approximations. For positive-term series, partial sums increase, so convergence is equivalent to being bounded above.

Conditional series require more care because cancellation can control partial sums even when the sum of magnitudes diverges. The definition through ordered partial sums also explains why rearranging conditionally convergent terms can change behavior.

Conceptual meaning

The ellipsis does not mean one literally completes infinitely many additions. It asks whether the sequence of finite totals settles. Convergence tests study that limiting behavior, often without producing the value of the sum.

A dependable method and decision rule

1. Write the general term and starting index clearly.
2. Check the necessary condition $a_n\to0$.
3. Recognize special sums such as geometric or telescoping forms.
4. Otherwise select a convergence test based on positivity, signs, powers, or factorials.
5. Separate the conclusion about convergence from any request to compute the sum.

Fully worked example

Graphical or geometric meaning

Plot $S_N$ rather than only $a_n$. Terms show step sizes; partial sums show location. A convergent series has step sizes tending to zero and positions approaching a horizontal level.

Common mistakes and why they fail

Verification and reasonableness checks

• Compute the first few partial sums from the claimed formula.
• Confirm convergence implies $a_n\to0$.
• Use an error or tail estimate when approximating a sum.

Partial sums are the objects that converge

The notation $\sum_{n=1}^\infty a_n$ means $\lim_{N\to\infty}S_N$ for $S_N=\sum_{n=1}^N a_n$. Terms may approach zero while partial sums fail to settle. Check $a_n\to0$ first, then choose a test from the structure: geometric ratio, positive comparison, alternating cancellation, factorial growth, or a related integral. A conclusion should name the test's hypotheses and result. Changing finitely many early terms alters the sum but not convergence, so eventual comparisons are enough. If convergence is established, distinguish an exact sum from a numerical approximation and use an error estimate whenever the chosen test supplies one. A convergence test rarely computes the sum automatically.

Practice

1. What are the first three partial sums of $\sum1/2^n$, starting at $n=1$?
2. If $S_N\to4$, what is the series sum?
3. If $a_n\to2$, can $\sum a_n$ converge?

Connections and next steps

• Sequences
• Geometric Series
• Divergence Test