Partial Sums

Principle

Partial sums turn an infinite series into a sequence of finite calculations. If those finite sums approach one stable finite value, the infinite series converges to that value; if they do not, the series diverges.

For physics models, partial sums are the practical approximations: use the first few contributions, then decide whether adding more terms is settling the result or driving it away.

Notation

• \(S_N\) is the partial sum using terms up to the index \(N\).
• \(a_n\) is the term with index \(n\).
• \(N\) is the number of included terms when the series starts at \(n=1\).
• \(\lim_{N\to\infty}S_N\) is the limiting value of the partial sums, if it exists.
• A series converges when \(S_N\) approaches a finite limit.
• A series diverges when \(S_N\) has no finite limit.

Method

To test an infinite series using partial sums, keep the terms and the partial sums separate.

1. Write the term formula \(a_n\).
2. Form the finite partial sum \(S_N\).
3. Study \(\lim_{N\to\infty}S_N\).
4. Before doing more work, check the necessary term condition: if \(a_n\) does not tend to \(0\), the series diverges.
5. If \(a_n\to0\), continue with a stronger argument; the term condition alone does not prove convergence.

Rules

• The partial sum is \(S_N=\sum_{n=1}^{N}a_n\) when the series starts at \(n=1\).
• Convergence means the sequence \(S_1,S_2,S_3,\ldots\) has a finite limit.
• If \(\lim_{n\to\infty}a_n\ne0\), then \(\sum a_n\) diverges.
• The condition \(a_n\to0\) is necessary for convergence but not sufficient.
• Infinite sums should not be rearranged unless a valid convergence result permits it.

Examples

For the arithmetic series \(\sum_{n=1}^{\infty}n\), the partial sums are explicit:

As \(N\) grows, \(N(N+1)/2\) grows without bound, so the arithmetic series diverges.

For the harmonic series \(\sum_{n=1}^{\infty}1/n\), the terms \(a_n=1/n\) tend to \(0\). That passes only the necessary term check. The partial sums still grow without a finite limiting value, so the harmonic series diverges.

The contrast is important: \(\sum n\) fails because the terms do not tend to zero, while \(\sum 1/n\) shows that terms tending to zero can still leave divergent partial sums.

Checks

• Check the limit of partial sums, not only the limit of the terms.
• Do not treat \(a_n\to0\) as proof of convergence.
• If \(a_n\) does not tend to \(0\), stop: the series diverges.
• Avoid rearranging the terms of an infinite series without a convergence reason.