Arithmetic
\(1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}\)
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Theory & Formula
A series is the sum of the terms of a sequence. Different types of series have different properties and formulas for their sums.
Common series types:
- Arithmetic: S_n = (n/2)(2a + (n-1)d) where a is first term, d is common difference
- Geometric: S_n = a(1-r^n)/(1-r) where a is first term, r is common ratio
- Infinite Geometric: S = a/(1-r) for |r| < 1 (convergent)
- Power Sums: ∑n = n(n+1)/2, ∑n² = n(n+1)(2n+1)/6
- Harmonic Series: ∑1/n diverges (grows without bound)
\(S_n = \sum_{i=1}^{n} a_i\)
Worked Examples
Geometric
\(1 + r + r^2 + \cdots = \frac{1}{1-r}\) for \(|r| < 1\)
Power Sum
\(1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}\)
Arithmetic Formula
\(S_n = \frac{n}{2}(2a + (n-1)d)\)