Calculus Examples

$\sum_{k = 1}^{3} \frac{1}{k^{2} + 4}$

Step 1

Expand the series for each value of $k$ .

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

Step 2

Simplify.

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Step 2.1

Find the common denominator.

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Step 2.1.1

Multiply $\frac{1}{1^{2} + 4}$ by $\frac{(2^{2} + 4) (3^{2} + 4)}{(2^{2} + 4) (3^{2} + 4)}$ .

$\frac{1}{1^{2} + 4} \cdot \frac{(2^{2} + 4) (3^{2} + 4)}{(2^{2} + 4) (3^{2} + 4)} + \frac{1}{2^{2} + 4} + \frac{1}{3^{2} + 4}$

Step 2.1.2

Multiply $\frac{1}{1^{2} + 4}$ by $\frac{(2^{2} + 4) (3^{2} + 4)}{(2^{2} + 4) (3^{2} + 4)}$ .

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

Step 2.1.3

Multiply $\frac{1}{2^{2} + 4}$ by $\frac{(1^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) (3^{2} + 4)}$ .

$\frac{(2^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) ((2^{2} + 4) (3^{2} + 4))} + \frac{1}{2^{2} + 4} \cdot \frac{(1^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) (3^{2} + 4)} + \frac{1}{3^{2} + 4}$

Step 2.1.4

Multiply $\frac{1}{2^{2} + 4}$ by $\frac{(1^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) (3^{2} + 4)}$ .

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

Step 2.1.5

Multiply $\frac{1}{3^{2} + 4}$ by $\frac{(1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4)}$ .

$\frac{(2^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) ((2^{2} + 4) (3^{2} + 4))} + \frac{(1^{2} + 4) (3^{2} + 4)}{(2^{2} + 4) ((1^{2} + 4) (3^{2} + 4))} + \frac{1}{3^{2} + 4} \cdot \frac{(1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4)}$

Step 2.1.6

Multiply $\frac{1}{3^{2} + 4}$ by $\frac{(1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4)}$ .

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

Step 2.1.7

Reorder the factors of $(1^{2} + 4) ((2^{2} + 4) (3^{2} + 4))$ .

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

Step 2.1.8

Reorder the factors of $(2^{2} + 4) ((1^{2} + 4) (3^{2} + 4))$ .

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

Step 2.1.9

Reorder the factors of $(3^{2} + 4) ((1^{2} + 4) (2^{2} + 4))$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Simplify each term.

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Step 2.3.1

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

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

Step 2.3.2

Add $4$ and $4$ .

$\frac{8 (3^{2} + 4) + (1^{2} + 4) (3^{2} + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.3.3

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

$\frac{8 (9 + 4) + (1^{2} + 4) (3^{2} + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.3.4

Add $9$ and $4$ .

$\frac{8 \cdot 13 + (1^{2} + 4) (3^{2} + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.3.5

Multiply $8$ by $13$ .

$\frac{104 + (1^{2} + 4) (3^{2} + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.3.6

One to any power is one.

$\frac{104 + (1 + 4) (3^{2} + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.3.7

Add $1$ and $4$ .

$\frac{104 + 5 (3^{2} + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.3.8

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

$\frac{104 + 5 (9 + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.3.9

Add $9$ and $4$ .

$\frac{104 + 5 \cdot 13 + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.3.10

Multiply $5$ by $13$ .

$\frac{104 + 65 + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.3.11

One to any power is one.

$\frac{104 + 65 + (1 + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.3.12

Add $1$ and $4$ .

$\frac{104 + 65 + 5 (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.3.13

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

$\frac{104 + 65 + 5 (4 + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.3.14

Add $4$ and $4$ .

$\frac{104 + 65 + 5 \cdot 8}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.3.15

Multiply $5$ by $8$ .

$\frac{104 + 65 + 40}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.4

Add $104$ and $65$ .

$\frac{169 + 40}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.5

Add $169$ and $40$ .

$\frac{209}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.6

Simplify the denominator.

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Step 2.6.1

One to any power is one.

$\frac{209}{(1 + 4) (2^{2} + 4) (3^{2} + 4)}$

Step 2.6.2

Add $1$ and $4$ .

$\frac{209}{5 (2^{2} + 4) (3^{2} + 4)}$

Step 2.6.3

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

$\frac{209}{5 (4 + 4) (3^{2} + 4)}$

Step 2.6.4

Add $4$ and $4$ .

$\frac{209}{5 \cdot 8 (3^{2} + 4)}$

Step 2.6.5

Multiply $5$ by $8$ .

$\frac{209}{40 (3^{2} + 4)}$

Step 2.6.6

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

$\frac{209}{40 (9 + 4)}$

Step 2.6.7

Add $9$ and $4$ .

$\frac{209}{40 \cdot 13}$

Step 2.7

Multiply $40$ by $13$ .

$\frac{209}{520}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{209}{520}$

Decimal Form:

$0.40192307 \dots$