Calculus Examples

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

Step 1

Expand the series for each value of $n$ .

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

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Subtract $1$ from $1$ .

$4 {(\frac{1}{4})}^{0} + 4 {(\frac{1}{4})}^{2 - 1} + 4 {(\frac{1}{4})}^{3 - 1} + 4 {(\frac{1}{4})}^{4 - 1}$

Step 2.1.2

Apply the product rule to $\frac{1}{4}$ .

$4 \frac{1^{0}}{4^{0}} + 4 {(\frac{1}{4})}^{2 - 1} + 4 {(\frac{1}{4})}^{3 - 1} + 4 {(\frac{1}{4})}^{4 - 1}$

Step 2.1.3

Anything raised to $0$ is $1$ .

$4 \frac{1}{4^{0}} + 4 {(\frac{1}{4})}^{2 - 1} + 4 {(\frac{1}{4})}^{3 - 1} + 4 {(\frac{1}{4})}^{4 - 1}$

Step 2.1.4

Anything raised to $0$ is $1$ .

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

Step 2.1.5

Cancel the common factor of $1$ .

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Step 2.1.5.1

Cancel the common factor.

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

Step 2.1.5.2

Rewrite the expression.

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

Step 2.1.6

Multiply $4$ by $1$ .

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

Step 2.1.7

Subtract $1$ from $2$ .

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

Step 2.1.8

Simplify.

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

Step 2.1.9

Cancel the common factor of $4$ .

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Step 2.1.9.1

Cancel the common factor.

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

Step 2.1.9.2

Rewrite the expression.

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

Step 2.1.10

Subtract $1$ from $3$ .

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

Step 2.1.11

Apply the product rule to $\frac{1}{4}$ .

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

Step 2.1.12

One to any power is one.

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

Step 2.1.13

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

$4 + 1 + 4 (\frac{1}{16}) + 4 {(\frac{1}{4})}^{4 - 1}$

Step 2.1.14

Cancel the common factor of $4$ .

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Step 2.1.14.1

Factor $4$ out of $16$ .

$4 + 1 + 4 \frac{1}{4 (4)} + 4 {(\frac{1}{4})}^{4 - 1}$

Step 2.1.14.2

Cancel the common factor.

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

Step 2.1.14.3

Rewrite the expression.

$4 + 1 + \frac{1}{4} + 4 {(\frac{1}{4})}^{4 - 1}$

Step 2.1.15

Subtract $1$ from $4$ .

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

Step 2.1.16

Apply the product rule to $\frac{1}{4}$ .

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

Step 2.1.17

One to any power is one.

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

Step 2.1.18

Raise $4$ to the power of $3$ .

$4 + 1 + \frac{1}{4} + 4 (\frac{1}{64})$

Step 2.1.19

Cancel the common factor of $4$ .

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Step 2.1.19.1

Factor $4$ out of $64$ .

$4 + 1 + \frac{1}{4} + 4 \frac{1}{4 (16)}$

Step 2.1.19.2

Cancel the common factor.

$4 + 1 + \frac{1}{4} + 4 \frac{1}{4 \cdot 16}$

Step 2.1.19.3

Rewrite the expression.

$4 + 1 + \frac{1}{4} + \frac{1}{16}$

Step 2.2

Find the common denominator.

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Step 2.2.1

Write $4$ as a fraction with denominator $1$ .

$\frac{4}{1} + 1 + \frac{1}{4} + \frac{1}{16}$

Step 2.2.2

Multiply $\frac{4}{1}$ by $\frac{16}{16}$ .

$\frac{4}{1} \cdot \frac{16}{16} + 1 + \frac{1}{4} + \frac{1}{16}$

Step 2.2.3

Multiply $\frac{4}{1}$ by $\frac{16}{16}$ .

$\frac{4 \cdot 16}{16} + 1 + \frac{1}{4} + \frac{1}{16}$

Step 2.2.4

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

$\frac{4 \cdot 16}{16} + \frac{1}{1} + \frac{1}{4} + \frac{1}{16}$

Step 2.2.5

Multiply $\frac{1}{1}$ by $\frac{16}{16}$ .

$\frac{4 \cdot 16}{16} + \frac{1}{1} \cdot \frac{16}{16} + \frac{1}{4} + \frac{1}{16}$

Step 2.2.6

Multiply $\frac{1}{1}$ by $\frac{16}{16}$ .

$\frac{4 \cdot 16}{16} + \frac{16}{16} + \frac{1}{4} + \frac{1}{16}$

Step 2.2.7

Multiply $\frac{1}{4}$ by $\frac{4}{4}$ .

$\frac{4 \cdot 16}{16} + \frac{16}{16} + \frac{1}{4} \cdot \frac{4}{4} + \frac{1}{16}$

Step 2.2.8

Multiply $\frac{1}{4}$ by $\frac{4}{4}$ .

$\frac{4 \cdot 16}{16} + \frac{16}{16} + \frac{4}{4 \cdot 4} + \frac{1}{16}$

Step 2.2.9

Multiply $4$ by $4$ .

$\frac{4 \cdot 16}{16} + \frac{16}{16} + \frac{4}{16} + \frac{1}{16}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{4 \cdot 16 + 16 + 4 + 1}{16}$

Step 2.4

Multiply $4$ by $16$ .

$\frac{64 + 16 + 4 + 1}{16}$

Step 2.5

Add $64$ and $16$ .

$\frac{80 + 4 + 1}{16}$

Step 2.6

Add $80$ and $4$ .

$\frac{84 + 1}{16}$

Step 2.7

Add $84$ and $1$ .

$\frac{85}{16}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{85}{16}$

Decimal Form:

$5.3125$

Mixed Number Form:

$5 \frac{5}{16}$