From a List of Terms to a Sequence of Sums
A sequence gives a term at each positive integer index. A series begins with such a sequence of terms, but asks a different question: what happens when we add more and more of those terms?
For example, given terms \(a_1,a_2,a_3,\ldots\), we can add the first term, then the first two terms, then the first three, and continue. These finite sums form a new sequence. Its behavior—not an imagined act of adding infinitely many numbers all at once—is what defines whether the series converges.
Partial Sums and Series Notation
Let \((a_n)\) be a sequence of real numbers. The nth partial sum is the sum of the first \(n\) terms. We denote it by \(s_n\):
The index \(k\) indicates which terms are being added, and \(n\) specifies where to stop. For each fixed \(n\), this is an ordinary finite sum. The values \(s_1,s_2,s_3,\ldots\) form the sequence of partial sums.
The infinite-series notation
refers to the series formed from the terms \(a_1,a_2,\ldots\). To determine whether this series has a finite sum, examine the partial sums \(s_n\).
There are therefore two related sequences to keep distinct: the sequence of terms \((a_n)\), and the sequence of partial sums \((s_n)\). The series’ convergence is defined using the second sequence.
Why the Partial-Sum Sequence Is the Key
An infinite series is not a finite list of additions with one last step. There is no final term at which to stop. Instead, we can ask a precise question about every finite stage: how much has been accumulated after \(n\) terms? This gives \(s_n\). We can then use the definition of a sequence limit to ask whether those accumulated amounts get arbitrarily close to one fixed finite number as \(n\) grows.
The link between consecutive partial sums follows directly from their definition. The sum through term \(n+1\) contains all the terms in the sum through term \(n\), plus one additional term, \(a_{n+1}\). Thus:
This relation helps construct partial sums one at a time. It also makes clear that a partial sum is an accumulated total, not merely the latest term.
Write down the sequence \(a_1,a_2,\ldots\) that defines the series.
For each \(n\), add the first \(n\) terms to obtain \(s_n\).
Ask whether \(s_n\) approaches a finite number as \(n\) tends to infinity.
If \(s_n\) has a finite limit, the series converges to that limit; otherwise, it diverges.
Worked Examples
Worked Example: A Series with a Zero Tail
Consider the series whose terms are \(a_1=2\), \(a_2=-3\), \(a_3=4\), and \(a_n=0\) for every \(n\geq4\). Find its partial sums and determine whether the series converges.
Compute the first partial sums directly:
For \(n\geq4\), each additional term is zero. Therefore \(s_4=s_3+0=3\), \(s_5=s_4+0=3\), and every later partial sum remains \(3\). In sequence notation, \(s_n=3\) for all \(n\geq3\), so:
Conclusion and check: The series converges to \(3\). The partial sums agree with the original finite nonzero terms: \(2-3+4=3\), and adding any remaining zero terms leaves that value unchanged.
Worked Example: Partial Sums That Oscillate
Consider the series with terms \(a_n=(-1)^{n+1}\), so the terms are \(1,-1,1,-1,\ldots\). Compute the partial sums and decide whether the series converges.
Adding terms successively gives:
The partial sums continue to alternate: \(s_n=1\) when \(n\) is odd and \(s_n=0\) when \(n\) is even. In particular, the subsequence of odd-indexed partial sums is constantly \(1\), while the subsequence of even-indexed partial sums is constantly \(0\). Those two subsequences have different limits, so the full sequence of partial sums cannot have a limit.
Conclusion and check: The series diverges. The computed pattern is consistent with \(s_{n+1}=s_n+a_{n+1}\): after a partial sum of \(1\), adding \(-1\) gives \(0\); after a partial sum of \(0\), adding \(1\) gives \(1\).
Worked Example: Partial Sums That Grow Without Bound
Consider the series with \(a_n=2\) for every positive integer \(n\). Find its partial-sum sequence and determine the series’ behavior.
The first few sums are:
There are \(n\) terms in \(s_n\), and each one equals \(2\), so \(s_n=2n\). As \(n\) increases, \(2n\) grows without bound; in particular, it does not approach any finite real number.
Conclusion and check: The series diverges. Substituting \(n=1,2,3,4\) into \(s_n=2n\) gives \(2,4,6,8\), which matches the direct finite sums.
Justification: A Series Sum Is a Limit
The definition of series convergence is built from the definition of a sequence limit. Here is the precise connection. Suppose the partial sums are \(s_n=\sum_{k=1}^{n}a_k\), and suppose the sequence \((s_n)\) has the finite limit \(S\). By the definition of a sequence limit, for every \(\varepsilon>0\), there is a positive integer \(N\) such that whenever \(n\geq N\),
This says that after sufficiently many terms have been included, the accumulated total is within any chosen positive tolerance of \(S\). Since \(s_n\) is exactly the total of the first \(n\) terms, this is precisely what it means for the series to converge to \(S\). Conversely, if the series is said to converge to \(S\), that statement means its partial sums satisfy this limit condition. Thus series convergence is neither a separate kind of limit nor an informal claim about adding forever: it is convergence of the sequence \((s_n)\).
A Common Mistake: Cancelling Terms Without Checking the Partial Sums
For the alternating series \(1-1+1-1+\cdots\), one might group neighboring terms and claim that each pair sums to zero, so the whole series must have sum zero. That conclusion is not justified. The partial sums are \(1,0,1,0,\ldots\), not a sequence approaching zero. The odd-indexed partial sums remain \(1\), while the even-indexed ones remain \(0\), so the partial sums have no limit.
Grouping or rearranging infinitely many terms requires care; a calculation on selected groups does not replace checking the sequence of partial sums in the stated order. For this series, the partial-sum definition gives the correct conclusion: it diverges.
Why This View Matters
The partial-sum definition gives a precise meaning to an infinite series before any tests or special formulas are used. It also distinguishes the question “What are the terms?” from the question “Do the accumulated totals settle toward a finite value?” Those questions concern different sequences, and answering the second requires studying partial sums.
When the terms follow a pattern, the partial sums may be easy to find; in other cases, their behavior can be harder to establish. In either situation, the definition stays the same. First identify the finite total \(s_n\); then determine whether those totals approach a finite limit. That is the foundation for every later method of analyzing series.
Check Your Understanding
Use the definition of partial sums and series convergence to answer these questions.
- Write the first four partial sums of the series with terms \(3,-1,2,-4,\ldots\).
- In the notation \(s_n=\sum_{k=1}^{n}a_k\), what does \(n\) specify?
- What limit must exist for a series to converge, and what does that limit represent?
- Explain why the series \(1,-1,1,-1,\ldots\) does not converge by referring to its partial sums.
- For the series with \(a_n=3\) for every \(n\), write a formula for \(s_n\) and state whether the series converges.
