Calculus Examples

\sum_{n = 0}^{\infty} {(\frac{1}{2})}^{n}

$\sum_{n = 0}^{\infty} {(\frac{1}{2})}^{n}$

Step 1

The sum of an infinite geometric series can be found using the formula

\frac{a}{1 - r}

$\frac{a}{1 - r}$ where

a

$a$ is the first term and

r

$r$ is the ratio between successive terms.

Step 2

Find the ratio of successive terms by plugging into the formula

r = \frac{a_{n + 1}}{a_{n}}

$r = \frac{a_{n + 1}}{a_{n}}$ and simplifying.

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Step 2.1

Substitute

a_{n}

$a_{n}$ and

a_{n + 1}

$a_{n + 1}$ into the formula for

r

$r$ .

r = \frac{{(\frac{1}{2})}^{n + 1}}{{(\frac{1}{2})}^{n}}

$r = \frac{{(\frac{1}{2})}^{n + 1}}{{(\frac{1}{2})}^{n}}$

Step 2.2

Cancel the common factor of

{(\frac{1}{2})}^{n + 1}

${(\frac{1}{2})}^{n + 1}$ and

{(\frac{1}{2})}^{n}

${(\frac{1}{2})}^{n}$ .

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Step 2.2.1

Factor

{(\frac{1}{2})}^{n}

${(\frac{1}{2})}^{n}$ out of

{(\frac{1}{2})}^{n + 1}

${(\frac{1}{2})}^{n + 1}$ .

r = \frac{{(\frac{1}{2})}^{n} \frac{1}{2}}{{(\frac{1}{2})}^{n}}

$r = \frac{{(\frac{1}{2})}^{n} \frac{1}{2}}{{(\frac{1}{2})}^{n}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Multiply by

1

$1$ .

r = \frac{{(\frac{1}{2})}^{n} \frac{1}{2}}{{(\frac{1}{2})}^{n} \cdot 1}

$r = \frac{{(\frac{1}{2})}^{n} \frac{1}{2}}{{(\frac{1}{2})}^{n} \cdot 1}$

Step 2.2.2.2

Cancel the common factor.

r = \frac{{(\frac{1}{2})}^{n} \frac{1}{2}}{{(\frac{1}{2})}^{n} \cdot 1}

$r = \frac{{(\frac{1}{2})}^{n} \frac{1}{2}}{{(\frac{1}{2})}^{n} \cdot 1}$

Step 2.2.2.3

Rewrite the expression.

r = \frac{\frac{1}{2}}{1}

$r = \frac{\frac{1}{2}}{1}$

Step 2.2.2.4

Divide

\frac{1}{2}

$\frac{1}{2}$ by

1

$1$ .

r = \frac{1}{2}

$r = \frac{1}{2}$

r = \frac{1}{2}

$r = \frac{1}{2}$

r = \frac{1}{2}

$r = \frac{1}{2}$

r = \frac{1}{2}

$r = \frac{1}{2}$

Step 3

Since

| r | < 1

$| r | < 1$ , the series converges.

Step 4

Find the first term in the series by substituting in the lower bound and simplifying.

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Step 4.1

Substitute

0

$0$ for

n

$n$ into

{(\frac{1}{2})}^{n}

${(\frac{1}{2})}^{n}$ .

a = {(\frac{1}{2})}^{0}

$a = {(\frac{1}{2})}^{0}$

Step 4.2

Simplify.

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Step 4.2.1

Apply the product rule to

\frac{1}{2}

$\frac{1}{2}$ .

a = \frac{1^{0}}{2^{0}}

$a = \frac{1^{0}}{2^{0}}$

Step 4.2.2

Anything raised to

0

$0$ is

1

$1$ .

a = \frac{1}{2^{0}}

$a = \frac{1}{2^{0}}$

Step 4.2.3

Anything raised to

0

$0$ is

1

$1$ .

a = \frac{1}{1}

$a = \frac{1}{1}$

Step 4.2.4

Divide

1

$1$ by

1

$1$ .

a = 1

$a = 1$

a = 1

$a = 1$

a = 1

$a = 1$

Step 5

Substitute the values of the ratio and first term into the sum formula.

\frac{1}{1 - \frac{1}{2}}

$\frac{1}{1 - \frac{1}{2}}$

Step 6

Simplify.

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Step 6.1

Simplify the denominator.

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Step 6.1.1

Write

1

$1$ as a fraction with a common denominator.

\frac{1}{\frac{2}{2} - \frac{1}{2}}

$\frac{1}{\frac{2}{2} - \frac{1}{2}}$

Step 6.1.2

Combine the numerators over the common denominator.

\frac{1}{\frac{2 - 1}{2}}

$\frac{1}{\frac{2 - 1}{2}}$

Step 6.1.3

Subtract

1

$1$ from

2

$2$ .

\frac{1}{\frac{1}{2}}

$\frac{1}{\frac{1}{2}}$

\frac{1}{\frac{1}{2}}

$\frac{1}{\frac{1}{2}}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

1 \cdot 2

$1 \cdot 2$

Step 6.3

Multiply

2

$2$ by

1

$1$ .

2

$2$

2

$2$