From Finite Sums to Infinite Series
An infinite series begins with a sequence of terms, but its meaning depends on more than the terms considered one at a time. To define what the terms add up to, we form longer and longer finite sums and ask whether those sums approach a finite limit.
This distinction matters in computation and in reasoning. A finite sum uses a specified number of terms. An infinite series is defined by the limiting behavior of all its partial sums. A finite partial sum may approximate the series value, but it is not automatically equal to that value.
Partial Sums and the Definition of a Series
Let \(a_1,a_2,a_3,\ldots\) be a sequence of real numbers. For each positive integer \(N\), its \(N\)th partial sum is the finite sum of its first \(N\) terms:
The sequence \(S_1,S_2,S_3,\ldots\) is called the sequence of partial sums. An infinite series is defined by asking whether this sequence has a finite limit as \(N\) tends to infinity.
The symbol \(\sum_{n=1}^{\infty}a_n\) indicates an infinite series. Its value is defined only when the partial sums converge. By contrast, \(\sum_{n=1}^{N}a_n\) is an ordinary finite sum for a particular \(N\); it is the partial sum \(S_N\), whether or not the infinite series converges.
Why the Limit Defines the Infinite Sum
The definition says that a convergent infinite series is not obtained by carrying out an endless addition in a final step. Instead, its value describes what the finite partial sums approach. The limit condition makes the idea of “approach” precise.
In other words, once enough terms have been included, every later partial sum is within any prescribed positive distance \(\varepsilon\) of \(S\). This is why a partial sum can serve as a finite approximation to the series value: for any chosen accuracy, convergence guarantees that sufficiently many terms produce an approximation within that accuracy.
For a positive integer \(N\), calculate \(S_N=a_1+\cdots+a_N\).
As \(N\) increases, the values \(S_N\) may approach a finite number, grow without bound, or fail to settle toward one value.
If \(S_N\to S\), the series converges to \(S\). Otherwise, it diverges and has no finite value under this definition.
The definition also explains why the infinite-series notation should not be read as “add a certain large but finite number of terms.” Any specific choice of \(N\) produces only \(S_N\). The infinite series concerns the behavior of the entire sequence of these finite sums.
Worked Examples
Worked Example: A Series with Only Three Nonzero Terms
Define \(a_1=3\), \(a_2=-2\), \(a_3=5\), and \(a_n=0\) for every \(n\geq4\). Find the value of \(\sum_{n=1}^{\infty}a_n\), if it exists.
First calculate the partial sums:
For every \(N\geq4\), all additional terms are zero, so
Thus \(S_N=6\) for every \(N\geq3\), and therefore \(\lim_{N\to\infty}S_N=6\). The series converges to \(6\). To check, every partial sum from \(S_3\) onward is exactly \(6\), so the partial sums do approach the stated value.
Worked Example: Alternating Terms That Do Not Produce a Limit
Consider \(a_n=(-1)^{n+1}\), whose terms are \(1,-1,1,-1,\ldots\). Determine whether its infinite series converges.
The first partial sums are
In general, for each positive integer \(k\), \(S_{2k}=0\) and \(S_{2k-1}=1\). If the sequence of partial sums had a limit \(L\), then both its even-indexed and odd-indexed subsequences would also approach \(L\). But the even-indexed partial sums are always \(0\), while the odd-indexed partial sums are always \(1\); they cannot approach the same number. Hence the partial sums have no limit, and the series diverges.
As a direct check, partial sums continue to alternate between \(0\) and \(1\), so they do not settle near a single finite value.
Worked Example: A Finite Partial Sum Is Not the Infinite Series
Consider the harmonic series \(\sum_{n=1}^{\infty}\frac1n\). Calculate its fourth partial sum, then determine whether the infinite series has a finite value.
The fourth partial sum is a finite sum:
This computes \(S_4\), not the value of the infinite series. To decide whether a finite series value exists, consider larger partial sums. For each positive integer \(m\), group the terms from \(2^{j-1}+1\) to \(2^j\), for \(j=1,\ldots,m\). In the \(j\)th group there are \(2^{j-1}\) terms, and each is at least \(1/2^j\). Therefore that group contributes at least \(2^{j-1}/2^j=1/2\). It follows that
As \(m\) increases, \(1+m/2\) grows without bound. Thus the partial sums \(S_N\) are unbounded: among them are the values \(S_{2^m}\), which exceed any fixed bound when \(m\) is large enough. A convergent sequence must be bounded, so these partial sums cannot converge to a finite limit. The series diverges. This check confirms that \(25/12\) is just the fourth partial sum, not a finite value for the infinite series.
A Common Mistake: Treating an Approximation as the Sum
A common error is to calculate a few terms and then label the result as the value of the infinite series. For instance, the calculation
is correct, but it states only that the fourth partial sum is \(25/12\). It does not establish that \(\sum_{n=1}^{\infty}\frac1n=25/12\). In fact, as the preceding example shows, the harmonic series diverges.
The correction is to name the finite calculation accurately and then address the limit separately: first write \(S_4=25/12\); then determine whether \(S_N\) has a finite limit as \(N\to\infty\). A finite calculation alone cannot settle that limit.
Why This Definition Matters
Defining series through partial sums gives a single framework for different behaviors. Partial sums may eventually become constant, may fail to settle, or may grow without bound. In each case, the question is the same: does the sequence \(S_N\) approach a finite limit?
This framework also keeps exact values separate from approximations. When a series converges to \(S\), a partial sum \(S_N\) can approximate \(S\), but it is generally a different number. When a series diverges, no finite number can be its value under the definition, even though every individual partial sum is finite.
Check Your Understanding
Answer these questions using the definition of an infinite series.
- For a sequence of terms \(a_n\), how is its \(N\)th partial sum \(S_N\) defined?
- What condition on the partial sums makes \(\sum_{n=1}^{\infty}a_n\) converge?
- If \(S_5=12\), what does this tell you, and what does it not establish about the infinite series?
- Why does the alternating series with terms \(1,-1,1,-1,\ldots\) diverge under the partial-sum definition?
- Can an infinite series have a finite value if its partial sums are unbounded? Explain using the definition.
