Defining Convergent and Divergent Infinite Series

3. Visual Connection

5. Concept Development

BC-only topic. An infinite series is not evaluated by completing infinitely many additions. It is defined by a limit of ordinary finite sums. This definition is the foundation for every convergence test in Unit 10.

Sequence terms versus a series

A sequence is an ordered list \(a_1,a_2,a_3,\ldots\). A series is the accumulation of those terms:

The symbol \(n\) is a dummy index. Changing its letter does not change the series, but changing the starting index can add or remove terms.

Partial sums make the definition precise

The \(N\)th partial sum is the finite sum of the first \(N\) terms:

The values \(S_1,S_2,S_3,\ldots\) form a new sequence called the sequence of partial sums. Do not confuse \(a_N\), the newest term, with \(S_N\), the total accumulated through that term.

Definition of convergence

The series converges only when the partial sums approach one finite real number. That limit is called the sum of the series. If the partial-sum limit does not exist as a finite number, the series diverges.

Different ways a series can diverge

• Growth to \(+\infty\): partial sums increase without bound.
• Growth to \(-\infty\): partial sums decrease without bound.
• Oscillation: partial sums alternate between separated values.
• Irregular behavior: partial sums remain unsettled without approaching one limit.

“Diverges” does not mean that every divergent series has sum infinity. Infinity is not a finite series sum.

Recover terms from partial sums

Since consecutive partial sums differ by the newest term,

This relationship allows a problem to specify a series indirectly through a formula, table, or graph for \(S_N\).

A necessary condition for convergence

If \(S_N\to S\), then

Therefore every convergent series must satisfy \(a_N\to0\). The converse is false: terms can approach zero too slowly for their accumulated partial sums to converge. Topic 10.3 develops this divergence test formally.

Telescoping partial sums

Some terms can be rewritten as differences \(a_n=b_n-b_{n+1}\). In a finite partial sum, most intermediate quantities cancel:

The infinite series is then decided by the limit of the remaining boundary expression, not by informal cancellation across an unfinished infinite sum.

Starting index and finite changes

Adding, deleting, or changing finitely many terms changes the numerical sum of a convergent series but does not change whether its tail converges. For an integer \(m>1\),

whenever the original series converges. This principle is useful when reindexing or isolating a tail.

Linearity for convergent series

If \(\sum a_n=A\) and \(\sum b_n=B\) both converge, then for constants \(c,d\),

The convergence assumptions matter. Algebraic combinations of divergent series cannot generally be manipulated as though they had finite sums.

Tables and graphs of partial sums

A table of \(S_N\) suggests convergence when the values settle toward one number, but a finite table alone is not proof. A graph of partial sums should be read as a discrete sequence: convergence means the plotted heights approach one horizontal level as \(N\) grows.

Partial sums as approximations

When a series converges to \(S\), \(S_N\) is a finite approximation and

is the remainder after \(N\) terms. This topic defines the approximation; later topics provide practical error bounds for specific kinds of series.

Definition-first checklist

1. Identify the term sequence \(a_n\) and starting index.
2. Write or determine the finite partial sum \(S_N\).
3. Simplify \(S_N\) before taking a limit.
4. Evaluate \(\lim\limits_{N\to\infty}S_N\).
5. Conclude “converges to \(S\)” only for a finite limit; otherwise state how it diverges.
6. Keep conclusions about \(a_n\) separate from conclusions about \(S_N\).

6. Detailed Worked Example and Error Check

Example 1: Convergence from an explicit partial sum

Suppose \(S_N=N/(N+1)\). Then

Therefore the series converges and its sum is \(1\).

Example 2: Oscillating partial sums

If \(S_N=(-1)^N\), then the even partial sums equal \(1\) and the odd partial sums equal \(-1\). The sequence has no limit, so the series diverges by oscillation.

Example 3: Unbounded partial sums

If \(S_N=\ln(N+1)\), then \(S_N\to+\infty\). The series diverges; it does not have “sum infinity” under the definition of convergence.

Example 4: A basic telescoping series

Use partial fractions:

The finite partial sum is

Thus \(S_N\to1\), so \(\sum_{n=1}^{\infty}1/[n(n+1)]=1\).

Example 5: Two uncanceled terms at each end

For

the partial sum is

Therefore the series converges to \(1/2+1/3=5/6\).

Example 6: Recover the terms from \(S_N\)

Let \(S_N=3-2/N\). The first term is \(a_1=S_1=1\). For \(N\ge2\),

The series sum is \(\lim S_N=3\). The separate formula for \(a_1\) is essential.

Example 7: Remove finitely many terms

Suppose \(\sum_{n=1}^{\infty}a_n=6\), with \(a_1=2\) and \(a_2=-1\). Then

Removing the first two terms changes the sum from \(6\) to \(5\), but the remaining tail still converges.

Example 8: Terms approach zero but the series diverges

For the harmonic series, \(a_n=1/n\to0\). However, grouping terms in partial sums at powers of two gives

These partial sums are unbounded, so \(\sum1/n\) diverges. The condition \(a_n\to0\) was necessary but not sufficient.

Example 9: Terms and partial sums both oscillate

For \(\sum_{n=1}^{\infty}(-1)^n\), the terms alternate between \(-1\) and \(1\), while

The partial sums do not settle, so the series diverges.

Example 10: Infer a partial-sum pattern from a table

Suppose the first partial sums are \(1/2,3/4,7/8,15/16\). The pattern

fits the table and has limit \(1\). If this formula is established for every \(N\), the series converges to \(1\). The table alone only suggests the pattern.

Common errors

• Calling \(a_N\) the partial sum.
• Treating the infinity symbol as a last index that can be substituted.
• Concluding convergence merely because \(a_N\to0\).
• Calling unbounded growth a finite sum.
• Inferring convergence from a short decimal table without a formula or theorem.
• Telescoping an infinite expression without first writing \(S_N\).
• Using \(a_N=S_N-S_{N-1}\) at \(N=1\) without defining \(S_0\).
• Forgetting that removing initial terms changes the sum even though it preserves convergence.
• Combining divergent series through ordinary sum laws.
• Confusing a sequence limit \(\lim a_n\) with the series sum \(\lim S_N\).