Series
Given a sequence \(\{a_k\}_{k=1}^\infty\) in a linear space \(V\), the expression \(\sum_{k=1}^\infty a_k = a_1 + a_2 + \cdots\) is called a series, and \(a_n\) is called the \(n\)-th term of this series. For a positive integer \(N\), \(\sum_{k=1}^N a_k = a_1 + \cdots + a_N\) is called the \(N\)-th partial sum of the series, denoted as \(S_N\).
If there exists an \(S \in V\) such that \(\lim_{N \to +\infty} S_N = S\), then the series \(\sum_{k=1}^\infty a_k\) is said to be convergent, and \(S\) is called the sum of the series \(\sum_{k=1}^\infty a_k\), denoted as \(S = \sum_{k=1}^\infty a_k\).
Here, the meaning of \(\lim_{N \to +\infty} S_N = S\) is: for any \(\varepsilon > 0\), there exists an \(N(\varepsilon)\) such that for any positive integer \(N > N(\varepsilon)\), we have \(\|S_N - S\| < \varepsilon\). Where \(\|\cdot\|\) is the norm in \(V\).
A series that does not converge is called a divergent series.
A series \(\sum_{k=1}^\infty a_k = a_1 + a_2 + \cdots\) is said to satisfy the Cauchy property if for any \(\varepsilon > 0\), there exists an \(N(\varepsilon)\) such that for any positive integers \(m > n > N(\varepsilon)\),
\[ \left\| \sum_{k=n}^m a_k \right\| < \varepsilon. \]
Theorem 8.1 (Cauchy Convergence Criterion) If a linear space \(V\) is complete under the norm \(\|\cdot\|\) (i.e., every Cauchy sequence converges), then the series \(\sum_{k=1}^\infty a_k = a_1 + a_2 + \cdots\) converges if and only if it satisfies the Cauchy property.
If the series \(\sum_{k=1}^\infty a_k\) converges, then:
\[ \lim_{n \to +\infty} \|a_n\| = 0. \]
Absolute Convergence and Comparison Tests
A series \(\sum_{k=1}^\infty a_k = a_1 + a_2 + \cdots\) is said to converge absolutely if the series of norms:
\[ \sum_{k=1}^\infty \|a_k\| \]
converges.
An absolutely convergent series is convergent.
Suppose \(a_k = O(b_k)\) as \(k \to +\infty\), that is, there exist \(K_0\) and \(M > 0\) such that \(\|a_k\| \le M\|b_k\|\) for all \(k \ge K_0\).
- If \(\sum_{k=1}^\infty \|b_k\|\) converges, then \(\sum_{k=1}^\infty \|a_k\|\) converges.
- If \(\sum_{k=1}^\infty \|a_k\|\) diverges, then \(\sum_{k=1}^\infty \|b_k\|\) diverges.
Theorem 8.2 (Integral Test) Let \(f: (0, +\infty) \to \mathbb{R}\) be a non-negative, monotonically decreasing function. Then the improper integral:
\[ \int_1^{+\infty} f(x) \, \mathrm{d}x \]
converges if and only if the series:
\[ \sum_{k=1}^\infty f(k) \]
converges.
Theorem 8.3 (D’Alembert’s Ratio Test) Let \(\sum_{k=1}^\infty a_k\) be a series in a complete normed linear space, and suppose \(a_k \neq 0\) for all sufficiently large \(k\). Let:
\[ R = \limsup_{k \to +\infty} \frac{\|a_{k+1}\|}{\|a_k\|}, \quad \text{and} \quad r = \liminf_{k \to +\infty} \frac{\|a_{k+1}\|}{\|a_k\|} \]
If \(R < 1\), the series \(\sum_{k=1}^\infty a_k\) converges absolutely.
If \(r > 1\), the series diverges.
If \(r \le 1 \le R\), the test is inconclusive.
Theorem 8.4 (Cauchy’s Root Test) Let \(\sum_{k=1}^\infty a_k\) be a series in a complete normed linear space. Let:
\[ \rho = \limsup_{k \to +\infty} \sqrt[k]{\|a_k\|} \]
- If \(\rho < 1\), the series \(\sum_{k=1}^\infty a_k\) converges absolutely.
- If \(\rho > 1\) (or \(\rho = +\infty\)), the series diverges.
- If \(\rho = 1\), the test is inconclusive.
Theorem 8.5 (Raabe’s Test) Let \(\sum_{k=1}^\infty a_k\) be a series in a complete normed linear space, and suppose \(a_k \neq 0\) for all sufficiently large \(k\). Let:
\[ \rho = \lim_{k \to +\infty} k \left( \frac{\|a_k\|}{\|a_{k+1}\|} - 1 \right) \]
If \(\rho > 1\), the series \(\sum_{k=1}^\infty a_k\) converges absolutely.
If \(\rho < 1\), the series of norms \(\sum_{k=1}^\infty \|a_k\|\) diverges (i.e., the series does not converge absolutely).
If \(\rho = 1\), the test is inconclusive.
Theorem 8.6 (Gauss’s Test) Let \(\sum_{k=1}^\infty a_k\) be a series in a complete normed linear space, and suppose \(a_k \neq 0\) for all sufficiently large \(k\). If the ratio of successive norms can be expressed as an asymptotic expansion of the form:
\[ \frac{\|a_k\|}{\|a_{k+1}\|} = 1 + \frac{\beta}{k} + O\left(\frac{1}{k^p}\right) \quad \text{as } k \to +\infty \]
where \(\beta\) is a real constant, and \(p > 1\) is a strictly positive constant (the remainder term is bounded by \(C/k^p\)). Then:
If \(\beta > 1\), the series \(\sum_{k=1}^\infty a_k\) converges absolutely.
If \(\beta \le 1\), the series of norms \(\sum_{k=1}^\infty \|a_k\|\) diverges.