Summation Notation in a Nutshell

Summation notation provides a compact way to represent the sum of a sequence of terms using the Greek capital sigma (Σ). You specify an index, its lower and upper bounds, and the general term (the “summand”) to be added.

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Notation Components

• Σ (Sigma): Indicates “sum of.”
• Index of summation: Often i, j, or k.
• Lower bound: Value at which the index starts (e.g., i=1).
• Upper bound: Value at which the index ends (e.g., i=n).
• Summand: Expression involving the index (e.g., i2 or ai).

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Common Examples

Summation	Expanded Form	Closed Form
Σi=1n i	1 + 2 + … + n	n(n + 1) / 2
Σi=1n i2	12 + 22 + … + n2	n(n + 1)(2n + 1) / 6
Σi=0n ri	1 + r + r2 + … + rn	(1 − rn+1) / (1 − r), if r ≠ 1

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Key Properties

• Linearity: Σ_i=a^b (c x_i + d y_i) = c Σ_i=a^b x_i + d Σ_i=a^b y_i.
• Index shift: Let k = i + m, then Σ_i=a^b f(i) = Σ_k=a+m^b+m f(k − m).
• Splitting sums: Σ_i=a^b x_i = Σ_i=a^c x_i + Σ_i=c+1^b x_i.
• Telescoping sums: If terms cancel pairwise, only boundary terms remain.

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Nested Summations

For double sums you nest two Σ’s. For example:

Σ_i=1^m Σ_j=1ⁿ a_ij

expands to:

a₁₁ + a₁₂ + … + a_1n + a₂₁ + … + a_mn.

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Beyond these basics, you might explore:

• How summation turns into an integral in the limit n → ∞.
• Connections with product notation (∏) and factorials.
• Generating functions built from power-series sums.
• Using summation in combinatorics (binomial sums, harmonic numbers).

Would you like to see a detailed proof of any closed-form formula or applications in a particular field?