Terms, Partial Sums, and Series Values

Three Objects to Keep Separate

A series involves three related but distinct objects: its terms, its partial sums, and, when one exists, its value. Keeping these separate prevents a common confusion: a list of terms is not itself the series’ value.

For example, the notation \(\sum_{n=1}^{\infty} a_n\) starts with a sequence of terms \(a_1,a_2,a_3,\ldots\). To ask for the value of this infinite series, we form sums of the first \(n\) terms and examine what happens as \(n\) increases.

Definition: Given terms \(a_1,a_2,a_3,\ldots\), the \(n\)th partial sum is \(s_n=a_1+a_2+\cdots+a_n=\sum_{k=1}^{n}a_k\). The infinite series \(\sum_{n=1}^{\infty}a_n\) has value \(S\) if the sequence of partial sums converges to the finite limit \(S\). In that case, we write \(\sum_{n=1}^{\infty}a_n=S\). If the partial sums do not converge to a finite limit, the series has no finite value.

Thus \(a_n\) is one term, \(s_n\) is a sum of the first \(n\) terms, and \(S\) is the limit of the partial sums when that limit exists. The subscript matters: \(a_n\) and \(s_n\) generally denote different sequences.

Finite Series: The Value Is the Last Partial Sum

A finite series has a final term. If its terms are \(a_1,\ldots,a_N\), its value is the finite sum \(a_1+\cdots+a_N\), which is also its last partial sum \(s_N\). There is no limit process needed to define that finite sum.

Worked Example: A Finite Series

Consider the four terms \(a_1=4\), \(a_2=-3\), \(a_3=2\), and \(a_4=-1\). First list the terms, then calculate each partial sum:

\begin{aligned} a_1&=4,\qquad a_2=-3,\qquad a_3=2,\qquad a_4=-1,\\ s_1&=a_1=4,\\ s_2&=a_1+a_2=4+(-3)=1,\\ s_3&=a_1+a_2+a_3=4+(-3)+2=3,\\ s_4&=a_1+a_2+a_3+a_4=4+(-3)+2+(-1)=2. \end{aligned}

The series value is its last partial sum, \(s_4=2\). Check the result by adding all four terms directly: \(4-3+2-1=2\). The terms are \(4,-3,2,-1\); the partial sums are \(4,1,3,2\); and the finite series value is \(2\).

A partial sum includes every term up to its index. For instance, \(s_3\) contains \(a_1,a_2,a_3\), not just \(a_3\). This cumulative structure is what connects the term sequence to the partial-sum sequence.

Why Consecutive Partial Sums Differ by One Term

The relationship between the two sequences can be stated exactly. Use the convention \(s_0=0\), representing the sum before any terms have been added. For \(n\geq 1\), the \(n\)th partial sum is the sum of the first \(n\) terms. Subtracting the preceding partial sum leaves only the newest term.

Theorem: For every \(n\geq 1\), \(s_n-s_{n-1}=a_n\). Equivalently, \(a_n=s_n-s_{n-1}\). Thus each term is the change from one partial sum to the next.

To justify this, write both partial sums as finite sums. When \(n\geq2\), \(s_n\) contains the first \(n-1\) terms and one additional term, while \(s_{n-1}\) contains just the first \(n-1\) terms. Subtracting cancels those shared terms. For \(n=1\), the convention \(s_0=0\) gives \(s_1-s_0=a_1-0=a_1\) as well.

\begin{aligned} s_n-s_{n-1} &=(a_1+a_2+\cdots+a_{n-1}+a_n) -(a_1+a_2+\cdots+a_{n-1})\\ &=a_n. \end{aligned}

This identity is a useful check on calculations: subtract consecutive partial sums and you should recover the term that was added. It also shows why the term sequence and partial-sum sequence are related without being identical.

Worked Example: Recovering Terms from Partial Sums

Suppose the partial sums begin \(s_0=0\), \(s_1=2\), \(s_2=-1\), and \(s_3=3\). The terms are the successive differences:

\begin{aligned} a_1&=s_1-s_0=2-0=2,\\ a_2&=s_2-s_1=-1-2=-3,\\ a_3&=s_3-s_2=3-(-1)=4. \end{aligned}

Check by building the partial sums from the recovered terms: \(2=2\), \(2+(-3)=-1\), and \(2+(-3)+4=3\). These agree with the given \(s_1,s_2,s_3\).

Infinite Series: A Value Requires a Limit

An infinite series has no last term, so it has no “last partial sum.” Its possible value is defined using the sequence \(s_1,s_2,s_3,\ldots\). If those partial sums approach a finite number \(S\), then \(S\) is the series value. If they do not approach any finite number, the series has no finite value.

This definition does not say that the terms themselves are the value, or that one can obtain the value by stopping after a few terms. A finite partial sum is a particular approximation; the infinite series value, when it exists, is the limit of all the partial sums.

Worked Example: An Infinite Series That Becomes Constant

Define an infinite sequence of terms by \(a_1=2\), \(a_2=-3\), \(a_3=4\), and \(a_n=0\) for every \(n\geq4\). The series continues indefinitely in its indexing, but all terms after the third are zero. Compute its partial sums:

\begin{aligned} s_1&=2,\\ s_2&=2+(-3)=-1,\\ s_3&=2+(-3)+4=3,\\ s_4&=s_3+a_4=3+0=3,\\ s_5&=s_4+a_5=3+0=3. \end{aligned}

For every \(n\geq3\), the partial sum remains \(3\), because each additional term is zero. Therefore \(s_n\) converges to \(3\), and the infinite series has value \(3\). Check the value from the terms: the only nonzero terms add to \(2-3+4=3\); every later addition leaves the sum unchanged.

This example is infinite in the sense that it has a term at every positive integer index. It also makes clear that “infinite series” does not mean that every term must be nonzero. What determines whether the series has a value is the behavior of its partial sums.

Infinite Series Without a Finite Value

Not every infinite series has a finite value. For example, if every term equals \(1\), the partial sums keep increasing. If terms alternate between \(1\) and \(-1\), the partial sums alternate between two different values. In both cases, we can see directly from the partial sums that there is no finite limit.

Worked Example: Partial Sums That Grow Without Bound

Let \(a_n=1\) for every \(n\geq1\). Each partial sum contains \(n\) copies of \(1\), so:

s_n=\underbrace{1+1+\cdots+1}_{n\text{ terms}}=n.

The first partial sums are \(s_1=1\), \(s_2=2\), \(s_3=3\), and \(s_4=4\). More generally, \(s_n=n\), which increases without bound as \(n\) increases. Hence the partial sums do not converge to a finite number, so the series \(\sum_{n=1}^{\infty}1\) has no finite value. The formula \(s_n=n\) checks directly: adding the next term gives \(s_{n+1}=n+1=s_n+1\), as required.

Worked Example: Partial Sums That Alternate

Let \(a_n=(-1)^{n+1}\), so the terms are \(1,-1,1,-1,\ldots\). The first partial sums are:

\begin{aligned} s_1&=1,\\ s_2&=1+(-1)=0,\\ s_3&=1+(-1)+1=1,\\ s_4&=1+(-1)+1+(-1)=0. \end{aligned}

For every positive integer \(k\), the first \(2k\) terms form \(k\) complete pairs. Each pair contributes \(1+(-1)=0\), so \(s_{2k}=0\). The next term is \(a_{2k+1}=1\), so \(s_{2k+1}=s_{2k}+1=1\). Thus the even-indexed partial sums are always \(0\), while the odd-indexed partial sums are always \(1\). Since these two subsequences have different limits, the full sequence of partial sums has no limit. The series therefore has no finite value.

A Common Mistake: Calling a Term the Series Value

A frequent error is to identify a series’ value with one of its terms, often the first term or the last term that has been calculated. This confuses the objects \(a_n\), \(s_n\), and \(S\). A term is one contribution; a partial sum combines a finite number of contributions; a series value is a limit of partial sums, if that limit exists.

Common mistake: For the terms \(2,-3,4,0,0,\ldots\), saying “the series value is \(4\) because the third term is \(4\)” is incorrect. The third term is \(a_3=4\), but the third partial sum is \(s_3=2+(-3)+4=3\), and the infinite series value is \(3\).

The correction is to label each quantity and calculate it according to its definition: identify the terms, add the first \(n\) terms to obtain \(s_n\), and only then determine whether the partial sums have a finite limit. If they do, that limit—not an individual term—is the series value.

Why the Distinction Matters

These distinctions provide the basic language for studying series. Questions about a series’ value are questions about the behavior of its partial-sum sequence. A few terms can help reveal a pattern, but the definition of an infinite series value requires a limit as the number of included terms grows without bound.

The term-to-partial-sum identity also gives a practical consistency check. If you have computed several partial sums, each difference \(s_n-s_{n-1}\) must equal the corresponding term \(a_n\). Conversely, if you know the partial sums, their consecutive differences recover the terms. These checks catch indexing and arithmetic errors before they affect a conclusion about the series.

Keep the three roles in view: the terms are the inputs being added, partial sums record the accumulated totals, and the series value is the finite limit of those totals when it exists. The alternating example shows that an infinite list of terms does not automatically have a value; the constant-partial-sum example shows how an infinite series can have one.

Check Your Understanding

For each question, distinguish the terms from the partial sums and from any series value.

For the finite terms \(5,-2,1\), calculate \(s_1,s_2,s_3\) and state the finite series value.
If \(s_0=0\), \(s_1=3\), and \(s_2=7\), what are \(a_1\) and \(a_2\)?
For \(a_n=0\) for every \(n\geq1\), write the partial sums and determine the infinite series value.
For the series with \(a_n=1\) for every \(n\), what is the formula for \(s_n\), and does the series have a finite value?
For the terms \(1,-1,1,-1,\ldots\), what are \(s_{2k}\) and \(s_{2k+1}\), and what do they imply about a series value?