Sequences and Series

Sigma Notation

Expand and evaluate a sum in sigma notation Σ_{k=klo}^{khi} f(k) by computing each term individually and adding them up.

Example

Expand sigma notation term by term and add the results.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

∑k=14(2k+1)\sum_{k= 1 }^{ 4 }( 2 k+ 1 )

Step 2 — k = 1

Evaluate the term at k = 1: get 3.

k=12⋅1+1=3k= 1 \quad 2 \cdot 1 + 1 = \hl{3}

Step 3 — k = 2

Evaluate the term at k = 2: get 5.

k=22⋅2+1=5k= 2 \quad 2 \cdot 2 + 1 = \hl{5}

Step 4 — k = 3

Evaluate the term at k = 3: get 7.

k=32⋅3+1=7k= 3 \quad 2 \cdot 3 + 1 = \hl{7}

Step 5 — k = 4

Evaluate the term at k = 4: get 9.

k=42⋅4+1=9k= 4 \quad 2 \cdot 4 + 1 = \hl{9}

Step 6 — Sum

Add the expanded terms to get 24.

3+5+7+9=243 + 5 + 7 + 9 = \hl{24}
sigma-notation Sigma notation Σ_{k=a}^{b} f(k) means: evaluate f at each integer k from a to b and sum the results. For Σ_{k=1}^{4} (2k+1): k=1: 3, k=2: 5, k=3: 7, k=4: 9 → total = 24. Each term is pinned exactly; the running sum verifies totals.