A series \(\sum_{n=1}^{\infty} a_n\) assigns a value to an infinite sum by means of partial sums. If the partial sums \(S_n = a_1 + a_2 + \cdots + a_n\) approach a finite limit as \(n \to \infty\), the series converges; otherwise, it diverges. Several standard series serve as benchmarks. The geometric series \(\sum_{n=0}^{\infty} a r^n\) converges to \(\frac{a}{1-r}\) whenever \(|r| < 1\) and diverges otherwise. The \(p\)-series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) converges when \(p > 1\) and diverges when \(p \leq 1\); the harmonic series \(\sum \frac{1}{n}\) is the borderline case that diverges. The Divergence Test gives a quick check: if \(\lim_{n \to \infty} a_n \neq 0\), the series must diverge, although the test is inconclusive when the limit is 0.
More powerful tests determine convergence in trickier cases. The Integral Test compares a series \(\sum a_n\) with the corresponding integral \(\int_1^\infty f(x) \, dx\), where \(a_n = f(n)\): if \(f\) is positive, continuous, and decreasing, then the series and the integral either both converge or both diverge. The Comparison Test uses known series for reference: if \(0 \leq a_n \leq b_n\) and \(\sum b_n\) converges, then \(\sum a_n\) converges. The Ratio Test computes \(L = \lim_{n \to \infty} |a_{n+1}/a_n|\); convergence is assured if \(L < 1\) and divergence if \(L > 1\), with \(L = 1\) inconclusive. The Root Test is similar, using \(L = \lim_{n \to \infty} |a_n|^{1/n}\) with the same thresholds. These last two are particularly suited to series involving factorials or \(n\)-th powers.
Alternating series, with terms that switch sign like \(\sum (-1)^n b_n\), are handled by the Alternating Series Test: such a series converges whenever \(b_n\) is decreasing and \(\lim_{n \to \infty} b_n = 0\). This makes possible conditional convergence, where \(\sum a_n\) converges but \(\sum |a_n|\) diverges (the alternating harmonic series \(\sum (-1)^{n+1}/n\) being a classic example, equal to \(\ln 2\)). Absolute convergence, where \(\sum |a_n|\) converges, is the stronger condition that always implies convergence. Distinguishing these two notions is essential in advanced analysis.
Taylor and Maclaurin series extend calculus to represent functions as infinite polynomials. The Taylor series of \(f\) centered at \(x = a\) is \[ \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n, \] and a Maclaurin series is simply the special case \(a = 0\). Among the most important are \(e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}\), \(\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}\), and \(\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}\), all convergent for every real \(x\). The radius of convergence \(R\) of a power series \(\sum a_n (x - c)^n\) is the threshold such that the series converges for \(|x - c| < R\) and diverges for \(|x - c| > R\); it can be found using the Ratio Test. These series provide a powerful bridge between functions and polynomials, enabling approximation and computation in settings where direct evaluation is impractical.