The Infinite Sum Paradox
A series is essentially the sum of the terms of a sequence. It asks the question: Can we add up infinitely many numbers and get a finite result? Consider the geometric series \(\sum_{n=1}^{\infty} \frac{1}{2^n} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots\)
Visualizing the Sum
We start with a unit square (Area = 1). At each step, we add a rectangle with area half of the remaining space.
Formal Definition
Given a sequence \((a_n)\), the infinite series \(\sum_{n=1}^{\infty} a_n\) is defined by the sequence of partial sums \((s_n)\), where: \[ s_n = \sum_{k=1}^{n} a_k = a_1 + a_2 + \dots + a_n \]
If \((s_n)\) does not converge, the series diverges.
Note: It is crucial to distinguish between the sequence of terms \((a_n)\) and the sequence of partial sums \((s_n)\). For the series to converge, the terms \(a_n\) must get smaller (in fact, \(\lim a_n = 0\) is necessary), but that alone is not sufficient.
Convergence Explorer
Compare how partial sums behave for different famous series.
Select a Series
The Harmonic Series diverges, even though the terms go to 0. It grows very slowly—logarithmically.
Cauchy Criterion for Series
Since a series converges if and only if its sequence of partial sums \((s_n)\) is a Cauchy sequence, we have:
Note that \(s_n - s_m = \sum_{k=m+1}^n a_k\). So the criterion says that the "tail" of the series can be made arbitrarily small.
Practice Problems
True/False
1. If \(\sum a_n\) converges, then \(\lim a_n = 0\).
True. This is the n-th term test for divergence (contrapositive).
2. If \(\lim a_n = 0\), then \(\sum a_n\) converges.
False. The harmonic series \(\sum \frac{1}{n}\) diverges even though \(\frac{1}{n} \to 0\).
3. The series \(\sum \frac{1}{n^p}\) converges if and only if \(p \ge 1\).
False. It converges if and only if \(p \gt 1\). For \(p=1\), it diverges.
4. If \(\sum |a_n|\) converges, then \(\sum a_n\) converges.
True. Absolute convergence implies convergence.
Fill in the Blank
1. A series \(\sum a_n\) converges if the sequence of ______ \((s_n = \sum_{k=1}^n a_k)\) converges.
Partial sums.
2. The geometric series \(\sum ar^n\) converges if and only if \(|r|\) is ______.
Less than 1 (or \( \lt 1\)).
3. The Cauchy Criterion for series states that \(\sum a_n\) converges if and only if for every \(\epsilon \gt 0\), there exists \(N\) such that for all \(n \ge m \gt N\), \(|\sum_{k=m}^n a_k| \lt \epsilon\). This is equivalent to saying the sequence of partial sums is a ______ sequence.
Cauchy.
Full Problems
1. Determine if \(\sum_{n=1}^\infty \frac{1}{\sqrt{n}}\) converges or diverges.
Diverges. This is a \(p\)-series with \(p = 1/2\). Since \(p \le 1\), the series diverges.
2. Calculate the sum of the series \(\sum_{n=0}^\infty 3(1/2)^n\).
This is a geometric series with \(a=3\) and \(r=1/2\). Since \(|r| \lt 1\), it converges to \(\frac{a}{1-r} = \frac{3}{1-1/2} = \frac{3}{1/2} = 6\).
3. Use the Cauchy Criterion to show that the harmonic series \(\sum \frac{1}{n}\) diverges.
Let \(\epsilon = 1/2\). For any \(N\), choose \(m=N+1\) and \(n=2N+2\). Then \(\sum_{k=m}^n \frac{1}{k} = \frac{1}{N+1} + \dots + \frac{1}{2N+2} \ge (N+2) \frac{1}{2N+2} = \frac{1}{2}\). Thus, the condition fails.
4. Prove that if \(\sum a_n\) converges, then \(\lim a_n = 0\).
Let \(s_n\) be the partial sums. Since \(\sum a_n\) converges, \(s_n \to S\) for some real number \(S\). Then \(s_{n-1} \to S\) as well. Since \(a_n = s_n - s_{n-1}\), we have \(\lim a_n = \lim s_n - \lim s_{n-1} = S - S = 0\).