An infinite series adds the terms of a sequence one after another, such as a1 + a2 + a3 + .... Some infinite sums settle toward a finite number, while others grow without bound. This matters because series are used to approximate functions, model repeated processes, and measure accumulated effects in physics, engineering, and probability.
The central idea is that infinitely many small pieces can have a finite total only if they shrink fast enough.
Key Facts
- An infinite series is a sum of sequence terms: Σ a_n = a_1 + a_2 + a_3 + ...
- A series converges if its partial sums S_N = a_1 + a_2 + ... + a_N approach a finite limit as N increases.
- A necessary condition for convergence is lim n→∞ a_n = 0, but this condition alone is not enough.
- The harmonic series diverges: Σ 1/n = 1 + 1/2 + 1/3 + ... grows without bound.
- A p-series Σ 1/n^p converges if p > 1 and diverges if p ≤ 1.
- For positive-term series, comparison tests show convergence when terms are smaller than a known convergent series and divergence when terms are larger than a known divergent series.
Vocabulary
- Infinite series
- An infinite series is the sum of all terms in an infinite sequence.
- Partial sum
- A partial sum is the sum of the first N terms of a series, written S_N.
- Convergence
- Convergence means the partial sums of a series approach one finite number.
- Divergence
- Divergence means the partial sums do not approach a finite limit.
- p-series
- A p-series is a series of the form Σ 1/n^p, whose convergence depends on the value of p.
Common Mistakes to Avoid
- Thinking terms going to zero guarantees convergence. This is wrong because the harmonic series has terms 1/n that approach zero, yet its total sum still diverges.
- Confusing a sequence with a series. A sequence lists values, while a series adds those values and studies the behavior of the partial sums.
- Using the p-series rule with the wrong exponent. The series Σ 1/n^p converges only when p > 1, so p = 1 is still divergent.
- Comparing only the first few terms of two series. Convergence depends on long-term behavior, so a valid comparison must work for sufficiently large n.
Practice Questions
- 1 Find the first four partial sums of the series Σ 1/2^n starting at n = 1, then predict whether the series converges and state its sum.
- 2 Decide whether each p-series converges or diverges: Σ 1/n^2, Σ 1/n, and Σ 1/n^0.5.
- 3 A series has positive terms that shrink toward zero, but each term is still larger than 1/n for all large n. Explain what this suggests about convergence or divergence.