Calculus Examples

$\sum_{k = 3}^{9} k^{2} + 1$

Step 1

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

$\sum_{k = 3}^{9} k^{2} + 1 = \sum_{k = 1}^{9} k^{2} + 1 - \sum_{k = 1}^{2} k^{2} + 1$

Step 2

Evaluate $\sum_{k = 1}^{9} k^{2} + 1$ .

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Step 2.1

Split the summation into smaller summations that fit the summation rules.

$\sum_{k = 1}^{9} k^{2} + 1 = \sum_{k = 1}^{9} k^{2} + \sum_{k = 1}^{9} 1$

Step 2.2

Evaluate $\sum_{k = 1}^{9} k^{2}$ .

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Step 2.2.1

The formula for the summation of a polynomial with degree $2$ is:

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

Step 2.2.2

Substitute the values into the formula.

$\frac{9 (9 + 1) (2 \cdot 9 + 1)}{6}$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Cancel the common factor of $9$ and $6$ .

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Step 2.2.3.1.1

Factor $3$ out of $9 (9 + 1) (2 \cdot 9 + 1)$ .

$\frac{3 (3 (9 + 1) (2 \cdot 9 + 1))}{6}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $3$ out of $6$ .

$\frac{3 (3 (9 + 1) (2 \cdot 9 + 1))}{3 (2)}$

Step 2.2.3.1.2.2

Cancel the common factor.

$\frac{3 (3 (9 + 1) (2 \cdot 9 + 1))}{3 \cdot 2}$

Step 2.2.3.1.2.3

Rewrite the expression.

$\frac{3 (9 + 1) (2 \cdot 9 + 1)}{2}$

Step 2.2.3.2

Simplify the numerator.

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Step 2.2.3.2.1

Multiply $2$ by $9$ .

$\frac{3 (9 + 1) (18 + 1)}{2}$

Step 2.2.3.2.2

Add $9$ and $1$ .

$\frac{3 \cdot 10 (18 + 1)}{2}$

Step 2.2.3.2.3

Multiply $3$ by $10$ .

$\frac{30 (18 + 1)}{2}$

Step 2.2.3.2.4

Add $18$ and $1$ .

$\frac{30 \cdot 19}{2}$

Step 2.2.3.3

Simplify the expression.

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Step 2.2.3.3.1

Multiply $30$ by $19$ .

$\frac{570}{2}$

Step 2.2.3.3.2

Divide $570$ by $2$ .

$285$

Step 2.3

Evaluate $\sum_{k = 1}^{9} 1$ .

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Step 2.3.1

The formula for the summation of a constant is:

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

Step 2.3.2

Substitute the values into the formula.

$(1) (9)$

Step 2.3.3

Multiply $9$ by $1$ .

$9$

Step 2.4

Add the results of the summations.

$285 + 9$

Step 2.5

Add $285$ and $9$ .

$294$

Step 3

Evaluate $\sum_{k = 1}^{2} k^{2} + 1$ .

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Step 3.1

Expand the series for each value of $k$ .

$1^{2} + 1 + 2^{2} + 1$

Step 3.2

Simplify.

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Step 3.2.1

Raise $1$ to the power of $2$ .

$1 + 1 + 2^{2} + 1$

Step 3.2.2

Add $1$ and $1$ .

$2 + 2^{2} + 1$

Step 3.2.3

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

$2 + 4 + 1$

Step 3.2.4

Add $4$ and $1$ .

$2 + 5$

Step 3.2.5

Add $2$ and $5$ .

$7$

Step 4

Replace the summations with the values found.

$294 - 7$

Step 5

Subtract $7$ from $294$ .

$287$