Divergence Test

The central idea

If $\sum a_n$ converges, then necessarily $a_n\to0$. Therefore, if $\lim a_n$ is nonzero or fails to exist, the series diverges. When the limit is zero, the divergence test is inconclusive; many convergent and divergent series share that term behavior.

Definitions, hypotheses, and notation

The theorem is often called the nth-term test for divergence to emphasize its one-way purpose. It should be checked first because it can end a problem quickly and because every stronger convergence test assumes, explicitly or implicitly, shrinking terms. A finite number of large early terms is harmless; the tail limit is decisive.

If the term limit is zero, inspect structure next: geometric ratios, $p$-series powers, factorials, alternating signs, or comparison targets. Writing 'the test fails' can be ambiguous; the accurate statement is that the test is inconclusive, not that the series converges.

Conceptual meaning

For partial sums $S_n$, the term $a_n=S_n-S_{n-1}$. If partial sums approach one finite limit, consecutive partial sums must become arbitrarily close, forcing $a_n$ to zero. Tiny additions are necessary but need not accumulate to a finite total.

A dependable method and decision rule

1. Identify the general term $a_n$, not the partial sum.
2. Compute $\lim_{n\to\infty}a_n$.
3. If the result is nonzero or nonexistent, conclude divergence immediately.
4. If the result is zero, write 'inconclusive' and choose another test.
5. Do not attempt to infer a sum from the term limit.

Fully worked example

Graphical or geometric meaning

If each late term remains near $3/2$, partial sums keep rising by roughly $1.5$ and cannot settle. When terms shrink to zero, a partial-sum graph may still drift without bound, just more slowly.

Common mistakes and why they fail

Verification and reasonableness checks

• Write the logical implication and its contrapositive.
• Compare with the harmonic series as a counterexample to the converse.
• Inspect whether the terms are even capable of becoming negligible.

A necessary condition is not a convergence test

If $\sum a_n$ converges, then $a_n\to0$. Therefore a nonzero limit, an infinite limit, or failure of the term limit proves divergence. The converse is false: $1/n\to0$ while the harmonic series diverges. Compute the term limit first because it may finish the problem cheaply; if it is zero, state that the divergence test is inconclusive and select another test. Do not confuse terms with partial sums. Convergence concerns $S_N=\sum_{n=1}^N a_n$, and the term condition follows from $a_N=S_N-S_{N-1}$. This distinction explains exactly why small terms can still accumulate without a finite total.

Practice

1. Test $\sum n/(n+1)$.
2. What does the test say about $\sum1/n^2$?
3. If $a_n$ oscillates between $1$ and $-1$, what happens?

Connections and next steps

• Infinite Series
• Harmonic Series
• Ratio Test