Series (Infinite Sum)

A series is the sum of the terms of a sequence. The finite series $\sum_{i=1}^n a_i = a_1 + a_2 + \cdots + a_n$ is just regular addition. The infinite series $\sum_{i=1}^\infty a_i$ is the limit of partial sums $S_n = \sum_{i=1}^n a_i$ as $n \to \infty$.

If $\lim_{n\to\infty} S_n$ exists and is finite, the series converges; otherwise it diverges. Famous examples:

• Geometric series $\sum r^n$ converges to $\frac{1}{1-r}$ when $|r| < 1$.
• Harmonic series $\sum \frac{1}{n}$ diverges (slowly).
• Basel problem: $\sum \frac{1}{n^2} = \frac{\pi^2}{6}$.

Convergence is decided by tests: ratio test, root test, integral test, comparison test, alternating-series test. Taylor series approximate functions as polynomials of arbitrarily high degree — the foundation of numerical analysis and physics approximations.