Sigma notation & sums

Japanese school year: Math B

What you learn

Learn the meaning of sigma notation and how to calculate various series using summation formulas. This notation is essential for expressing complicated sums concisely and preparing for calculus and limits. Prior understanding of arithmetic and geometric series will help you grasp these techniques smoothly.

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Key points

Represents adding a constant value $c$ for $n$ times from $k = 1$ to $n$ , which simply equals $nc$ . Use this when the term inside $\sum$ does not contain $k$ .

\sum_{k=1}^n c = nc

Formula to calculate the sum of consecutive integers from $1$ up to $n$ : $1 + 2 + \dots + n$ . This is the most fundamental summation formula for sequences.

\sum_{k=1}^n k = \frac{1}{2}n(n + 1)

Formula to calculate the sum of squares from $1$ up to $n$ : $1^2 + 2^2 + \dots + n^2$ . It is used when evaluating sums with quadratic terms.

\sum_{k=1}^n k^2 = \frac{1}{6}n(n + 1)(2n + 1)

Formula for the sum of cubes from $1$ up to $n$ ( $1^3 + 2^3 + \dots + n^3$ ). It equals the square of the ordinary sum $1 + 2 + \dots + n$ .

\sum_{k=1}^n k^3 = \left\{\frac{1}{2}n(n + 1)\right\}^2

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Practice 1Coming soon