Calculus Examples

$\sum_{n = 1}^{4} n^{- 2} (12)$

Step 1

Simplify the summation.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{n^{2}} \cdot 12$

Step 1.2

Combine $\frac{1}{n^{2}}$ and $12$ .

$\frac{12}{n^{2}}$

Step 1.3

Rewrite the summation.

$\sum_{n = 1}^{4} \frac{12}{n^{2}}$

Step 2

Expand the series for each value of $n$ .

$\frac{12}{1^{2}} + \frac{12}{2^{2}} + \frac{12}{3^{2}} + \frac{12}{4^{2}}$

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

One to any power is one.

$\frac{12}{1} + \frac{12}{2^{2}} + \frac{12}{3^{2}} + \frac{12}{4^{2}}$

Step 3.1.2

Divide $12$ by $1$ .

$12 + \frac{12}{2^{2}} + \frac{12}{3^{2}} + \frac{12}{4^{2}}$

Step 3.1.3

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

$12 + \frac{12}{4} + \frac{12}{3^{2}} + \frac{12}{4^{2}}$

Step 3.1.4

Divide $12$ by $4$ .

$12 + 3 + \frac{12}{3^{2}} + \frac{12}{4^{2}}$

Step 3.1.5

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

$12 + 3 + \frac{12}{9} + \frac{12}{4^{2}}$

Step 3.1.6

Cancel the common factor of $12$ and $9$ .

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Step 3.1.6.1

Factor $3$ out of $12$ .

$12 + 3 + \frac{3 (4)}{9} + \frac{12}{4^{2}}$

Step 3.1.6.2

Cancel the common factors.

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Step 3.1.6.2.1

Factor $3$ out of $9$ .

$12 + 3 + \frac{3 \cdot 4}{3 \cdot 3} + \frac{12}{4^{2}}$

Step 3.1.6.2.2

Cancel the common factor.

$12 + 3 + \frac{3 \cdot 4}{3 \cdot 3} + \frac{12}{4^{2}}$

Step 3.1.6.2.3

Rewrite the expression.

$12 + 3 + \frac{4}{3} + \frac{12}{4^{2}}$

Step 3.1.7

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

$12 + 3 + \frac{4}{3} + \frac{12}{16}$

Step 3.1.8

Cancel the common factor of $12$ and $16$ .

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Step 3.1.8.1

Factor $4$ out of $12$ .

$12 + 3 + \frac{4}{3} + \frac{4 (3)}{16}$

Step 3.1.8.2

Cancel the common factors.

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Step 3.1.8.2.1

Factor $4$ out of $16$ .

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

Step 3.1.8.2.2

Cancel the common factor.

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

Step 3.1.8.2.3

Rewrite the expression.

$12 + 3 + \frac{4}{3} + \frac{3}{4}$

Step 3.2

Find the common denominator.

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Step 3.2.1

Write $12$ as a fraction with denominator $1$ .

$\frac{12}{1} + 3 + \frac{4}{3} + \frac{3}{4}$

Step 3.2.2

Multiply $\frac{12}{1}$ by $\frac{12}{12}$ .

$\frac{12}{1} \cdot \frac{12}{12} + 3 + \frac{4}{3} + \frac{3}{4}$

Step 3.2.3

Multiply $\frac{12}{1}$ by $\frac{12}{12}$ .

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

Step 3.2.4

Write $3$ as a fraction with denominator $1$ .

$\frac{12 \cdot 12}{12} + \frac{3}{1} + \frac{4}{3} + \frac{3}{4}$

Step 3.2.5

Multiply $\frac{3}{1}$ by $\frac{12}{12}$ .

$\frac{12 \cdot 12}{12} + \frac{3}{1} \cdot \frac{12}{12} + \frac{4}{3} + \frac{3}{4}$

Step 3.2.6

Multiply $\frac{3}{1}$ by $\frac{12}{12}$ .

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

Step 3.2.7

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

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

Step 3.2.8

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

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

Step 3.2.9

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

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

Step 3.2.10

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

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

Step 3.2.11

Multiply $3$ by $4$ .

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

Step 3.2.12

Reorder the factors of $4 \cdot 3$ .

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

Step 3.2.13

Multiply $3$ by $4$ .

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 3.4

Simplify each term.

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Step 3.4.1

Multiply $12$ by $12$ .

$\frac{144 + 3 \cdot 12 + 4 \cdot 4 + 3 \cdot 3}{12}$

Step 3.4.2

Multiply $3$ by $12$ .

$\frac{144 + 36 + 4 \cdot 4 + 3 \cdot 3}{12}$

Step 3.4.3

Multiply $4$ by $4$ .

$\frac{144 + 36 + 16 + 3 \cdot 3}{12}$

Step 3.4.4

Multiply $3$ by $3$ .

$\frac{144 + 36 + 16 + 9}{12}$

Step 3.5

Add $144$ and $36$ .

$\frac{180 + 16 + 9}{12}$

Step 3.6

Add $180$ and $16$ .

$\frac{196 + 9}{12}$

Step 3.7

Add $196$ and $9$ .

$\frac{205}{12}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{205}{12}$

Decimal Form:

$17.08\overline{3}$

Mixed Number Form:

$17 \frac{1}{12}$