Summation Notation and Series Properties
Master sigma notation, index manipulation, and essential summation formulas.
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Questions Covered in This Set
10 cards to master
What does the sigma symbol Σ represent in mathematics?
The sum of a sequence of terms
In the notation ∑(i=m to n) aᵢ, what is 'i' called?
The index of summation (a dummy variable that changes with each term)
State the Constant Multiple Rule for summations.
∑(i=m to n) c·aᵢ = c·∑(i=m to n) aᵢ — you can factor constants out of a sum
What is the formula for the sum of the first n integers?
∑(i=1 to n) i = n(n+1)/2
What is the formula for the sum of the first n squares?
∑(i=1 to n) i² = n(n+1)(2n+1)/6
State the Sum/Difference Rule for summations.
∑(i=m to n) (aᵢ ± bᵢ) = ∑(i=m to n) aᵢ ± ∑(i=m to n) bᵢ — you can split sums/differences into separate sums
What is the sum of a constant c added n times: ∑(i=1 to n) c?
nc (the constant multiplied by the number of terms)
What is the formula for the sum of the first n cubes?
∑(i=1 to n) i³ = [n(n+1)/2]² — the square of the sum of first n integers
When shifting the index of summation, what must remain unchanged?
The actual terms being added must remain the same; only the indexing changes
How can you split a summation ∑(i=m to n) aᵢ into two parts?
∑(i=m to n) aᵢ = ∑(i=m to k) aᵢ + ∑(i=k+1 to n) aᵢ, where m ≤ k < n
