Calculus Examples

$\sum_{i = 2}^{8} i^{3}$

Step 1

Split the summation to make the starting value of $i$ equal to $1$ .

$\sum_{k = 2}^{8} i^{3} = \sum_{k = 1}^{8} i^{3} - \sum_{k = 1}^{1} i^{3}$

Step 2

Evaluate $\sum_{k = 1}^{8} i^{3}$ .

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Step 2.1

The formula for the summation of a polynomial with degree $3$ is:

$\sum_{k = 1}^{n} i^{3} = \frac{n^{2} {(n + 1)}^{2}}{4}$

Step 2.2

Substitute the values into the formula.

$\frac{8^{2} {(8 + 1)}^{2}}{4}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Add $8$ and $1$ .

$\frac{8^{2} \cdot 9^{2}}{4}$

Step 2.3.1.2

Raise $8$ to the power of $2$ .

$\frac{64 \cdot 9^{2}}{4}$

Step 2.3.1.3

Raise $9$ to the power of $2$ .

$\frac{64 \cdot 81}{4}$

Step 2.3.2

Simplify the expression.

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Step 2.3.2.1

Multiply $64$ by $81$ .

$\frac{5184}{4}$

Step 2.3.2.2

Divide $5184$ by $4$ .

$1296$

Step 3

Evaluate $\sum_{k = 1}^{1} i^{3}$ .

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Step 3.1

Expand the series for each value of $i$ .

$1^{3}$

Step 3.2

Raise $1$ to the power of $3$ .

$1$

Step 4

Replace the summations with the values found.

$1296 - 1$

Step 5

Subtract $1$ from $1296$ .

$1295$