Calculus Examples

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

Step 1

Expand the series for each value of $k$ .

$\frac{1^{2}}{1 + 1} + \frac{2^{2}}{2 + 1} + \frac{3^{2}}{3 + 1}$

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

One to any power is one.

$\frac{1}{1 + 1} + \frac{2^{2}}{2 + 1} + \frac{3^{2}}{3 + 1}$

Step 2.1.2

Add $1$ and $1$ .

$\frac{1}{2} + \frac{2^{2}}{2 + 1} + \frac{3^{2}}{3 + 1}$

Step 2.1.3

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

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

Step 2.1.4

Add $2$ and $1$ .

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

Step 2.1.5

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

$\frac{1}{2} + \frac{4}{3} + \frac{9}{3 + 1}$

Step 2.1.6

Add $3$ and $1$ .

$\frac{1}{2} + \frac{4}{3} + \frac{9}{4}$

Step 2.2

Find the common denominator.

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Step 2.2.1

Multiply $\frac{1}{2}$ by $\frac{6}{6}$ .

$\frac{1}{2} \cdot \frac{6}{6} + \frac{4}{3} + \frac{9}{4}$

Step 2.2.2

Multiply $\frac{1}{2}$ by $\frac{6}{6}$ .

$\frac{6}{2 \cdot 6} + \frac{4}{3} + \frac{9}{4}$

Step 2.2.3

Multiply $\frac{4}{3}$ by $\frac{4}{4}$ .

$\frac{6}{2 \cdot 6} + \frac{4}{3} \cdot \frac{4}{4} + \frac{9}{4}$

Step 2.2.4

Multiply $\frac{4}{3}$ by $\frac{4}{4}$ .

$\frac{6}{2 \cdot 6} + \frac{4 \cdot 4}{3 \cdot 4} + \frac{9}{4}$

Step 2.2.5

Multiply $\frac{9}{4}$ by $\frac{3}{3}$ .

$\frac{6}{2 \cdot 6} + \frac{4 \cdot 4}{3 \cdot 4} + \frac{9}{4} \cdot \frac{3}{3}$

Step 2.2.6

Multiply $\frac{9}{4}$ by $\frac{3}{3}$ .

$\frac{6}{2 \cdot 6} + \frac{4 \cdot 4}{3 \cdot 4} + \frac{9 \cdot 3}{4 \cdot 3}$

Step 2.2.7

Multiply $2$ by $6$ .

$\frac{6}{12} + \frac{4 \cdot 4}{3 \cdot 4} + \frac{9 \cdot 3}{4 \cdot 3}$

Step 2.2.8

Multiply $3$ by $4$ .

$\frac{6}{12} + \frac{4 \cdot 4}{12} + \frac{9 \cdot 3}{4 \cdot 3}$

Step 2.2.9

Reorder the factors of $4 \cdot 3$ .

$\frac{6}{12} + \frac{4 \cdot 4}{12} + \frac{9 \cdot 3}{3 \cdot 4}$

Step 2.2.10

Multiply $3$ by $4$ .

$\frac{6}{12} + \frac{4 \cdot 4}{12} + \frac{9 \cdot 3}{12}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{6 + 4 \cdot 4 + 9 \cdot 3}{12}$

Step 2.4

Simplify each term.

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Step 2.4.1

Multiply $4$ by $4$ .

$\frac{6 + 16 + 9 \cdot 3}{12}$

Step 2.4.2

Multiply $9$ by $3$ .

$\frac{6 + 16 + 27}{12}$

Step 2.5

Add $6$ and $16$ .

$\frac{22 + 27}{12}$

Step 2.6

Add $22$ and $27$ .

$\frac{49}{12}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{49}{12}$

Decimal Form:

$4.08\overline{3}$

Mixed Number Form:

$4 \frac{1}{12}$