A series adds the terms of a sequence. For the sequence 1, 2, 3, 4, the series is 1 + 2 + 3 + 4 = 10. Series can be finite or infinite. An infinite series may converge to a limit or diverge.

The geometric series 1 + 1/2 + 1/4 + 1/8 + ... converges to 2. The harmonic series 1 + 1/2 + 1/3 + ... diverges, even though its terms approach zero. Convergence tests, like the ratio test and comparison test, determine whether a series has a finite sum.

Series appear in calculus, physics, and engineering. Taylor series express functions as infinite sums. Fourier series decompose periodic functions into sine and cosine terms. Power series solve differential equations. The concept is central to analysis.

• Sum of terms of a sequence
• Can be finite or infinite
• Convergent or divergent
• Used in Taylor and Fourier expansions