Sequences and Series infographic - Convergence, Tests & Sums

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Calculus

Sequences and Series

Convergence, Tests & Sums

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Sequences and series are tools for describing what happens when a process continues step by step forever. A sequence lists numbers in order, while a series adds those numbers together. Convergence tells us whether the terms of a sequence approach a single value or whether the partial sums of a series settle toward a finite number. This idea matters because infinite processes appear in calculus, physics, engineering, and numerical computation.

To study convergence, mathematicians compare terms or partial sums to a limit and ask whether they get arbitrarily close after enough steps. For sequences, the key question is whether ana_n approaches LL as nn becomes large. For series, we examine the sequence of partial sums Sn=a1+a2+…+anS_n = a_1 + a_2 + \ldots + a_n and test whether SnS_n approaches a finite value. Different convergence tests, such as the geometric series rule, comparison test, ratio test, and integral test, help determine what happens without adding infinitely many terms directly.

Key Facts

  • A sequence converges to L if for every epsilon > 0, there exists N such that n > N implies |a_n - L| < epsilon.
  • A series ∑an\sum a_n converges if its partial sums Sn=∑k=1nakS_n = \sum_{k=1}^{n} a_k approach a finite limit.
  • If ∑an\sum a_n converges, then an→0a_n \to 0. The converse is false.
  • Geometric series: ∑n=0∞arn\sum_{n=0}^{\infty} ar^n converges to a1−r\frac{a}{1 - r} when ∣r∣<1|r| < 1.
  • p-series: ∑n=1∞1np\sum_{n=1}^{\infty} \frac{1}{n^p} converges if p>1p > 1 and diverges if p≤1p \leq 1.
  • Ratio test: if lim⁡n→∞∣an+1an∣=r\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = r, then the series converges for r<1r < 1 and diverges for r>1r > 1.

Vocabulary

Sequence
A sequence is an ordered list of numbers written as a_1, a_2, a_3, and so on.
Series
A series is the sum of the terms of a sequence, often written as a_1 + a_2 + a_3 + ...
Limit
A limit is the value that sequence terms or partial sums approach as the index grows without bound.
Partial sum
A partial sum is the sum of the first n terms of a series, usually denoted S_n.
Convergence test
A convergence test is a method used to decide whether an infinite series converges or diverges.

Common Mistakes to Avoid

  • Assuming an→0a_n \to 0 guarantees that ∑an\sum a_n converges, but this is wrong because many series, such as ∑1n\sum \frac{1}{n}, have terms going to zero and still diverge.
  • Applying a sequence limit directly to a series, which is wrong because a series must be studied through its partial sums rather than through the terms alone.
  • Using the geometric series formula when |r| >= 1, which is wrong because the infinite geometric series only converges for |r| < 1.
  • Stopping after the ratio test gives r = 1, which is wrong because the ratio test is inconclusive there and another test must be used.

Practice Questions

  1. 1 Determine whether the sequence a_n = (3n + 1)/(n + 5) converges, and if it does, find its limit.
  2. 2 Decide whether the series ∑n=1∞4(13)n\sum_{n=1}^{\infty} 4\left(\frac{1}{3}\right)^n converges, and if so, find its sum.
  3. 3 A student says that because the terms of the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... get very small, the series must converge. Explain clearly why this reasoning is not sufficient.