Calculus Examples

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

Step 1

Expand the series for each value of $k$ .

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

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Add $1$ and $1$ .

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

Step 2.1.2

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

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

Step 2.1.3

One to any power is one.

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

Step 2.1.4

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

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

Step 2.1.5

Add $2$ and $1$ .

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

Step 2.1.6

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

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

Step 2.1.7

One to any power is one.

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

Step 2.1.8

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

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

Step 2.1.9

Add $3$ and $1$ .

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

Step 2.1.10

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

$\frac{1}{4} + \frac{1}{8} + \frac{1^{4}}{2^{4}}$

Step 2.1.11

One to any power is one.

$\frac{1}{4} + \frac{1}{8} + \frac{1}{2^{4}}$

Step 2.1.12

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

$\frac{1}{4} + \frac{1}{8} + \frac{1}{16}$

Step 2.2

Find the common denominator.

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Step 2.2.1

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

$\frac{1}{4} \cdot \frac{4}{4} + \frac{1}{8} + \frac{1}{16}$

Step 2.2.2

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

$\frac{4}{4 \cdot 4} + \frac{1}{8} + \frac{1}{16}$

Step 2.2.3

Multiply $\frac{1}{8}$ by $\frac{2}{2}$ .

$\frac{4}{4 \cdot 4} + \frac{1}{8} \cdot \frac{2}{2} + \frac{1}{16}$

Step 2.2.4

Multiply $\frac{1}{8}$ by $\frac{2}{2}$ .

$\frac{4}{4 \cdot 4} + \frac{2}{8 \cdot 2} + \frac{1}{16}$

Step 2.2.5

Multiply $4$ by $4$ .

$\frac{4}{16} + \frac{2}{8 \cdot 2} + \frac{1}{16}$

Step 2.2.6

Reorder the factors of $8 \cdot 2$ .

$\frac{4}{16} + \frac{2}{2 \cdot 8} + \frac{1}{16}$

Step 2.2.7

Multiply $2$ by $8$ .

$\frac{4}{16} + \frac{2}{16} + \frac{1}{16}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{4 + 2 + 1}{16}$

Step 2.4

Add $4$ and $2$ .

$\frac{6 + 1}{16}$

Step 2.5

Add $6$ and $1$ .

$\frac{7}{16}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{16}$

Decimal Form:

$0.4375$