Calculus Examples

$\sum_{n = 1}^{4} 3 {(\frac{1}{10})}^{n}$

Step 1

Expand the series for each value of $n$ .

$3 {(\frac{1}{10})}^{1} + 3 {(\frac{1}{10})}^{2} + 3 {(\frac{1}{10})}^{3} + 3 {(\frac{1}{10})}^{4}$

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Simplify.

$3 (\frac{1}{10}) + 3 {(\frac{1}{10})}^{2} + 3 {(\frac{1}{10})}^{3} + 3 {(\frac{1}{10})}^{4}$

Step 2.1.2

Combine $3$ and $\frac{1}{10}$ .

$\frac{3}{10} + 3 {(\frac{1}{10})}^{2} + 3 {(\frac{1}{10})}^{3} + 3 {(\frac{1}{10})}^{4}$

Step 2.1.3

Apply the product rule to $\frac{1}{10}$ .

$\frac{3}{10} + 3 \frac{1^{2}}{10^{2}} + 3 {(\frac{1}{10})}^{3} + 3 {(\frac{1}{10})}^{4}$

Step 2.1.4

One to any power is one.

$\frac{3}{10} + 3 \frac{1}{10^{2}} + 3 {(\frac{1}{10})}^{3} + 3 {(\frac{1}{10})}^{4}$

Step 2.1.5

Raise $10$ to the power of $2$ .

$\frac{3}{10} + 3 (\frac{1}{100}) + 3 {(\frac{1}{10})}^{3} + 3 {(\frac{1}{10})}^{4}$

Step 2.1.6

Combine $3$ and $\frac{1}{100}$ .

$\frac{3}{10} + \frac{3}{100} + 3 {(\frac{1}{10})}^{3} + 3 {(\frac{1}{10})}^{4}$

Step 2.1.7

Apply the product rule to $\frac{1}{10}$ .

$\frac{3}{10} + \frac{3}{100} + 3 \frac{1^{3}}{10^{3}} + 3 {(\frac{1}{10})}^{4}$

Step 2.1.8

One to any power is one.

$\frac{3}{10} + \frac{3}{100} + 3 \frac{1}{10^{3}} + 3 {(\frac{1}{10})}^{4}$

Step 2.1.9

Raise $10$ to the power of $3$ .

$\frac{3}{10} + \frac{3}{100} + 3 (\frac{1}{1000}) + 3 {(\frac{1}{10})}^{4}$

Step 2.1.10

Combine $3$ and $\frac{1}{1000}$ .

$\frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + 3 {(\frac{1}{10})}^{4}$

Step 2.1.11

Apply the product rule to $\frac{1}{10}$ .

$\frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + 3 \frac{1^{4}}{10^{4}}$

Step 2.1.12

One to any power is one.

$\frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + 3 \frac{1}{10^{4}}$

Step 2.1.13

Raise $10$ to the power of $4$ .

$\frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + 3 (\frac{1}{10000})$

Step 2.1.14

Combine $3$ and $\frac{1}{10000}$ .

$\frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \frac{3}{10000}$

Step 2.2

Find the common denominator.

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Step 2.2.1

Multiply $\frac{3}{10}$ by $\frac{1000}{1000}$ .

$\frac{3}{10} \cdot \frac{1000}{1000} + \frac{3}{100} + \frac{3}{1000} + \frac{3}{10000}$

Step 2.2.2

Multiply $\frac{3}{10}$ by $\frac{1000}{1000}$ .

$\frac{3 \cdot 1000}{10 \cdot 1000} + \frac{3}{100} + \frac{3}{1000} + \frac{3}{10000}$

Step 2.2.3

Multiply $\frac{3}{100}$ by $\frac{100}{100}$ .

$\frac{3 \cdot 1000}{10 \cdot 1000} + \frac{3}{100} \cdot \frac{100}{100} + \frac{3}{1000} + \frac{3}{10000}$

Step 2.2.4

Multiply $\frac{3}{100}$ by $\frac{100}{100}$ .

$\frac{3 \cdot 1000}{10 \cdot 1000} + \frac{3 \cdot 100}{100 \cdot 100} + \frac{3}{1000} + \frac{3}{10000}$

Step 2.2.5

Multiply $\frac{3}{1000}$ by $\frac{10}{10}$ .

$\frac{3 \cdot 1000}{10 \cdot 1000} + \frac{3 \cdot 100}{100 \cdot 100} + \frac{3}{1000} \cdot \frac{10}{10} + \frac{3}{10000}$

Step 2.2.6

Multiply $\frac{3}{1000}$ by $\frac{10}{10}$ .

$\frac{3 \cdot 1000}{10 \cdot 1000} + \frac{3 \cdot 100}{100 \cdot 100} + \frac{3 \cdot 10}{1000 \cdot 10} + \frac{3}{10000}$

Step 2.2.7

Multiply $10$ by $1000$ .

$\frac{3 \cdot 1000}{10000} + \frac{3 \cdot 100}{100 \cdot 100} + \frac{3 \cdot 10}{1000 \cdot 10} + \frac{3}{10000}$

Step 2.2.8

Multiply $100$ by $100$ .

$\frac{3 \cdot 1000}{10000} + \frac{3 \cdot 100}{10000} + \frac{3 \cdot 10}{1000 \cdot 10} + \frac{3}{10000}$

Step 2.2.9

Reorder the factors of $1000 \cdot 10$ .

$\frac{3 \cdot 1000}{10000} + \frac{3 \cdot 100}{10000} + \frac{3 \cdot 10}{10 \cdot 1000} + \frac{3}{10000}$

Step 2.2.10

Multiply $10$ by $1000$ .

$\frac{3 \cdot 1000}{10000} + \frac{3 \cdot 100}{10000} + \frac{3 \cdot 10}{10000} + \frac{3}{10000}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{3 \cdot 1000 + 3 \cdot 100 + 3 \cdot 10 + 3}{10000}$

Step 2.4

Simplify each term.

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Step 2.4.1

Multiply $3$ by $1000$ .

$\frac{3000 + 3 \cdot 100 + 3 \cdot 10 + 3}{10000}$

Step 2.4.2

Multiply $3$ by $100$ .

$\frac{3000 + 300 + 3 \cdot 10 + 3}{10000}$

Step 2.4.3

Multiply $3$ by $10$ .

$\frac{3000 + 300 + 30 + 3}{10000}$

Step 2.5

Add $3000$ and $300$ .

$\frac{3300 + 30 + 3}{10000}$

Step 2.6

Add $3300$ and $30$ .

$\frac{3330 + 3}{10000}$

Step 2.7

Add $3330$ and $3$ .

$\frac{3333}{10000}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{3333}{10000}$

Decimal Form:

$0.3333$