(464) Discrete Math - 2.4.3 Summations and Sigma Notation

The expression starts with the Sigma symbol (Σ), which indicates summation.
Each summation has components:
- Index (i): This denotes which term in the sequence is being considered.
- Terms (a sub i): Represents the values in the sequence being summed.
Limits of Summation:
- Lower Limit (M): The starting index (where i begins).
- Upper Limit (N): The index at which summation stops.
The different expressions of Sigma notation:
- ( \sum_{i=M}^{N} a_{i} )
- ( a_{M} + a_{M+1} + \ldots + a_{N} )
- All forms convey the same meaning: summing a sub i from M to N.

Consider the sequence defined by ( a_{i} = \frac{1}{i} )
- Terms: ( a_{1} = 1, a_{2} = \frac{1}{2}, a_{3} = \frac{1}{3}, \ldots, a_{100} = \frac{1}{100} )
The summation to express this sequence:
- Sigma notation: ( \sum_{i=1}^{100} \frac{1}{i} )
- This indicates summing all terms from 1 to 100 (i.e., ( 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{100} )).

Problem 1: Find the summation from 5 to 9 of ( i^2 ).
- Calculate each term: 25 (5^2), 36 (6^2), 49 (7^2), 64 (8^2), 81 (9^2).
- Sum: ( 25 + 36 + 49 + 64 + 81 = 255 )
Problem 2: Find the summation from 7 to 10 of ( (-1)^i ).
- Terms: ( (-1)^7 = -1, (-1)^8 = 1, (-1)^9 = -1, (-1)^{10} = 1 )
- Sum: ( -1 + 1 - 1 + 1 = 0 )

Review important properties and formulas pertaining to summations for calculations and proofs.