Why Partial Sums Matter
A series is written as an infinite addition, but its value is not found by adding infinitely many terms all at once. Instead, we examine a sequence of finite sums. That sequence reveals whether the running totals settle toward a finite number, remain fixed after some point, grow without bound, or keep oscillating.
This distinction matters because the terms of a series and its partial sums are different sequences. The terms describe what is added at each step; the partial sums record the accumulated result. It is the behavior of the partial sums that defines whether the series converges.
Terms and Partial Sums
Let \(a_1,a_2,a_3,\ldots\) be a sequence of real numbers. The infinite series formed from these terms is written \(\sum_{k=1}^{\infty}a_k\). For each positive integer \(n\), its \(n\)th partial sum is the finite sum of the first \(n\) terms.
The sequence \(a_1,a_2,\ldots\) is the sequence of terms. The sequence \(S_1,S_2,\ldots\) is the sequence of partial sums. The series is defined through the second sequence, not by treating the terms themselves as a total.
The index \(n\) tells us how many terms have been included. For example, \(S_3\) is the sum of the first three terms, not the third term alone. Increasing \(n\) by one adds exactly one new term:
This recurrence is a useful check on calculations: consecutive partial sums must differ by the term just added.
Worked Example: A Series Whose Partial Sums Stabilize
Worked Example: A Series Whose Partial Sums Stabilize
Consider the terms \(a_1=3\), \(a_2=-1\), \(a_3=4\), and \(a_n=0\) for every \(n\geq4\). Find the partial sums and the sum of the series.
Compute the first three partial sums:
Every later term is zero. Therefore, for \(n\geq3\), adding the next term leaves the total unchanged:
and the same holds for every later partial sum. Thus \(S_n=6\) for all \(n\geq3\), so \(\lim_{n\to\infty}S_n=6\). The series converges to \(6\).
Check: The first three terms add to \(3-1+4=6\), and all remaining terms are zero, so every partial sum from \(S_3\) onward is indeed \(6\).
This example stabilizes: from some index onward, all partial sums have exactly the same value. Stabilization guarantees convergence, because a sequence that is eventually equal to \(6\) has limit \(6\). But stabilization is not required for convergence. Partial sums can keep changing and still approach a finite limit.
Worked Example: Partial Sums That Approach a Limit
Worked Example: Partial Sums That Approach a Limit
We can construct a series by first specifying partial sums that approach \(5\), then defining each term as the change from one partial sum to the next. Set \(S_0=0\), and for \(n\geq1\) set
Define \(a_1=S_1\), and for \(n\geq2\) define \(a_n=S_n-S_{n-1}\). The first partial sums are
Consequently, the first terms are
For \(n\geq2\), the general term follows by subtracting the two partial-sum expressions:
By the way the terms were defined, the partial sum through \(n\) is
Every intermediate \(S_j\) for \(1\leq j<n\) appears once positively and once negatively, leaving only \(S_n\). Finally, since \(1/(n+1)\to0\), the displayed formula for \(S_n\) gives \(\lim_{n\to\infty}S_n=5\). Thus this series converges to \(5\), even though its partial sums do not become constant.
Check: At \(n=3\), the terms found above give \(a_1+a_2+a_3=\frac92+\frac56-\frac{7}{12}=\frac{54+10-7}{12}=\frac{57}{12}=\frac{19}{4}=S_3\), as required.
Why the Partial-Sum Limit Defines the Sum
The finite partial sums are what we can actually calculate. If the totals \(S_n\) approach \(L\), then taking more and more terms makes the running total as close to \(L\) as desired. This is precisely the meaning of saying that the infinite series has sum \(L\). The definition does not claim that an infinite list of additions is performed in one final step; it assigns the series the limit approached by its finite sums.
This statement is the definition of series convergence expressed as a testable question about a sequence. To determine convergence, calculate or characterize \(S_n\), then examine the limit of \(S_n\). A finite list of partial sums can suggest a pattern, but the convergence claim requires describing what happens as \(n\) grows without bound.
Worked Example: Partial Sums That Grow Without Bound
Worked Example: Partial Sums That Grow Without Bound
Consider the series whose every term is \(a_n=2\). Its \(n\)th partial sum is
These partial sums grow without bound. To verify that the series does not converge to a finite real number, suppose instead that \(S_n\to L\) for some real \(L\). By the definition of convergence, using \(\varepsilon=1\), there must be an index \(N\) such that whenever \(n\geq N\),
In particular, \(S_n<L+1\) for every \(n\geq N\). Choose an integer \(n\) large enough to satisfy both \(n\geq N\) and \(n>(L+1)/2\). Such an integer exists. For this choice, \(S_n=2n>L+1\), contradicting the required inequality \(S_n<L+1\). Therefore the partial sums cannot converge to any finite \(L\), and the series diverges.
Check: The formula \(S_n=2n\) agrees with the first values \(S_1=2\), \(S_2=4\), and \(S_3=6\); each new partial sum is the previous one plus \(2\).
The choice of \(n\) in this argument must satisfy the convergence condition \(n\geq N\) as well as the inequality needed for a contradiction. Merely choosing \(n\) large enough for the second inequality would not ensure that the convergence bound applies.
Worked Example: Partial Sums That Oscillate
Worked Example: Partial Sums That Oscillate
Let \(a_n=(-1)^{n+1}\), so the terms are \(1,-1,1,-1,\ldots\). Calculate the first partial sums:
Continuing this pattern, \(S_n=1\) when \(n\) is odd and \(S_n=0\) when \(n\) is even. The odd-indexed partial sums are constantly \(1\), while the even-indexed partial sums are constantly \(0\). If the full sequence \((S_n)\) had a limit, each of these subsequences would have the same limit. Since their limits are \(1\) and \(0\), respectively, the full sequence has no limit. The series diverges.
Check: Adding the next term alternates the total between \(1\) and \(0\): an odd partial sum of \(1\) becomes \(0\), and an even partial sum of \(0\) becomes \(1\). This confirms the stated pattern for all \(n\).
Two Ways Partial Sums Can Fail to Converge
The examples show two distinct kinds of divergence. In the constant-term example, the partial sums increase without bound. In the alternating example, they remain bounded but keep switching between different values. A sequence of partial sums can fail to have a finite limit in either way, and simply observing that its values are bounded is not enough to establish convergence.
| Partial-sum behavior | Conclusion for the series |
|---|---|
| The partial sums become equal to one fixed value and stay there | The series converges to that value |
| The partial sums keep changing but approach a finite limit | The series converges to that limit |
| The partial sums grow without bound | The series diverges |
| The partial sums oscillate without approaching one value | The series diverges |
Common Mistake: Confusing a Term with a Partial Sum
For the terms \(3,-1,4,0,0,\ldots\), it is incorrect to say that the partial sums are just the terms \(3,-1,4,0,\ldots\). That mistake would produce the false calculation \(S_2=-1\). The definition instead requires adding all terms through the current index:
In particular, \(S_2=S_1+a_2=3+(-1)=2\), which checks the result using the recurrence for consecutive partial sums. The term \(a_2=-1\) is the new amount added; the partial sum \(S_2=2\) is the accumulated total.
Why This Distinction Is Useful
Thinking in partial sums gives a precise meaning to an infinite series and a practical way to investigate it. Each finite calculation builds the sequence \(S_1,S_2,\ldots\), and the behavior of that sequence determines the series’ status. A series may settle immediately after a finite number of nonzero terms, approach a limit gradually, grow without bound, or fail to settle because it oscillates.
When examining a series, keep the two sequences separate: first identify the terms \(a_n\), then form the partial sums \(S_n\). State whether the partial sums have a finite limit, and if they do, give that limit as the series’ sum. This interpretation is the foundation for deciding what an infinite series means.
Check Your Understanding
For each question, focus on the partial-sum sequence rather than treating the terms as the total.
- For the terms \(a_1=5\), \(a_2=-2\), and \(a_n=0\) for \(n\geq3\), find \(S_1,S_2,S_3\), and state the series’ sum.
- If \(a_n=3\) for every positive integer \(n\), write a formula for \(S_n\). Does the series converge to a finite real number?
- What condition on the sequence of partial sums defines convergence of \(\sum_{k=1}^{\infty}a_k\) to \(L\)?
- Can partial sums approach a limit without ever becoming constant? Give the constructed example in this tutorial as evidence.
- For the terms \(1,-1,1,-1,\ldots\), identify the odd-indexed and even-indexed partial sums. What do they show about convergence?
