Calculus Examples

$\sum_{k = 1}^{4} k (k + 1)$

Step 1

Simplify the summation.

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Step 1.1

Apply the distributive property.

$k \cdot k + k \cdot 1$

Step 1.2

Multiply $k$ by $k$ .

$k^{2} + k \cdot 1$

Step 1.3

Multiply $k$ by $1$ .

$k^{2} + k$

Step 1.4

Rewrite the summation.

$\sum_{k = 1}^{4} k^{2} + k$

$\sum_{k = 1}^{4} k^{2} + k$

Step 2

Expand the series for each value of $k$ .

$1^{2} + 1 + 2^{2} + 2 + 3^{2} + 3 + 4^{2} + 4$

Step 3

Simplify.

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Step 3.1

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

$1 + 1 + 2^{2} + 2 + 3^{2} + 3 + 4^{2} + 4$

Step 3.2

Add $1$ and $1$ .

$2 + 2^{2} + 2 + 3^{2} + 3 + 4^{2} + 4$

Step 3.3

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

$2 + 4 + 2 + 3^{2} + 3 + 4^{2} + 4$

Step 3.4

Add $4$ and $2$ .

$2 + 6 + 3^{2} + 3 + 4^{2} + 4$

Step 3.5

Add $2$ and $6$ .

$8 + 3^{2} + 3 + 4^{2} + 4$

Step 3.6

Raise $3$ to the power of $2$ .

$8 + 9 + 3 + 4^{2} + 4$

Step 3.7

Add $9$ and $3$ .

$8 + 12 + 4^{2} + 4$

Step 3.8

Add $8$ and $12$ .

$20 + 4^{2} + 4$

Step 3.9

Raise $4$ to the power of $2$ .

$20 + 16 + 4$

Step 3.10

Add $16$ and $4$ .

$20 + 20$

Step 3.11

Add $20$ and $20$ .

$40$

$40$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$