Summation Notation

Context: FIT1058_MOC · the $\sum$ sign for adding a sequence’s terms · partial sums form a new sequence · evaluated by Arithmetic Series / Geometric Series

Quick Revision

• 🎯 Objective: ${\sum }_{i=1}^n{s}_i$ = add a sequence’s terms ➔ partial sums $(S_n)$ form a new sequence.
• 📦 Core Components: index ➔ range ➔ summand ➔ bound local variable.
• ⚡ Key Constraint: math fixes no order (associative/commutative); code’s finite arithmetic is order-sensitive.

📝 Core

1. The Notation

• Definition ➔ $S_n={\sum }_{i=1}^n{s}_i=s_1+\cdots +s_n$; partial sums are a new sequence.
• Anatomy ➔ $i$ = index; $i=1$ initial; $n$ final; $[1,n{]}_{\mathbb{N}}$ range; $s_i$ summand.
• Evaluate ➔ substitute each index value, add.

2. Notation vs Algorithm

• Math ➔ specifies the result, fixes no order (addition associative + commutative).
• Program ➔ also fixes order, names/initialises the total; finite arithmetic isn’t perfectly associative ⟹ rounding depends on order.

3. The Index Is Bound

• Arbitrary name ➔ ${\sum }_i{s}_i={\sum }_j{s}_j$ (renamed consistently).
• Local ➔ scope = the summation, like a bound variable under a quantifier.
• Reuse warning ➔ outer $i$ differs from summation’s $i$.

Key identities:

$S_n={\sum }_{i=1}^n{s}_i=s_1+s_2+\cdots +s_n,{\sum }_{i=1}^n{s}_i={\sum }_{j=1}^n{s}_j(\text{reindex})$

⚖️ Core Decision Matrix

Aspect	Notation	Program
specifies	the result	result + order
order	none (assoc/comm)	fixed
arithmetic	exact	finite precision
rounding	n/a	order-dependent

When It Flips: studying the partial-sum sequence $(S_n)$ is often the goal; its closed form comes from arithmetic / geometric series formulas. Mathematically the sum is unique; in floating-point, order changes accumulated error.

📊 Exam Execution Trace

Manual Execution Trace

${\sum }_{i=1}^5(2i-1)$:

Step / State	$i$	$s_i=2i-1$	$S_i$
0 (Init)	—	—	0
1	1	1	1
2	2	3	4
3	3	5	9
4	4,5	7,9	16,25

⚠️ Common Mistakes

• 💡 Reindex everywhere ➔ change both limits and the summand together, and avoid a name already used outside; the index is a bound, local variable.

🧠 Active Recall

Name every part of ${\sum }_{i=1}^n{s}_i$ and why the letter $i$ doesn't matter.

• Hint: Bound local index.

Answer

• Short answer: $i$ index, $i=1$ initial, $n$ final, $[1,n]$ range, $s_i$ summand; renaming consistently leaves the value unchanged.
• Why: Scope ➔ the index is local, like a quantifier-bound variable.

Why does summation impose no order, yet a program's sum can be order-dependent?

• Hint: Exact vs finite precision.

Answer

• Short answer: Addition is associative/commutative, so $\sum$ is a single value; a program fixes order and uses finite precision.
• Why: Rounding ➔ different orders accumulate error differently.