Calculus Examples

$\sum_{i = 1}^{3} 4 x {(\frac{1}{2})}^{i - 1}$

Step 1

Expand the series for each value of $i$ .

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

Step 2

Simplify.

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Step 2.1

Subtract $1$ from $1$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 2.2.2

Anything raised to $0$ is $1$ .

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

Step 2.2.3

Anything raised to $0$ is $1$ .

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

Step 2.2.4

Divide $1$ by $1$ .

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

Step 2.2.5

Multiply $4$ by $1$ .

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

Step 2.2.6

Subtract $1$ from $2$ .

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

Step 2.2.7

Simplify.

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

Step 2.2.8

Cancel the common factor of $2$ .

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Step 2.2.8.1

Factor $2$ out of $4 x$ .

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

Step 2.2.8.2

Cancel the common factor.

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

Step 2.2.8.3

Rewrite the expression.

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

Step 2.2.9

Subtract $1$ from $3$ .

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

Step 2.2.10

Apply the product rule to $\frac{1}{2}$ .

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

Step 2.2.11

One to any power is one.

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

Step 2.2.12

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

$4 x + 2 x + 4 x \frac{1}{4}$

Step 2.2.13

Cancel the common factor of $4$ .

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Step 2.2.13.1

Factor $4$ out of $4 x$ .

$4 x + 2 x + 4 (x) \frac{1}{4}$

Step 2.2.13.2

Cancel the common factor.

$4 x + 2 x + 4 x \frac{1}{4}$

Step 2.2.13.3

Rewrite the expression.

$4 x + 2 x + x$

Step 2.3

Add $4 x$ and $2 x$ .

$6 x + x$

Step 2.4

Add $6 x$ and $x$ .

$7 x$