Calculus Examples

$\sum_{i = 1}^{\infty} 5 \cdot {0.5}^{i - 1}$

Step 1

The sum of an infinite geometric series can be found using the formula $\frac{a}{1 - r}$ where $a$ is the first term and $r$ is the ratio between successive terms.

Step 2

Find the ratio of successive terms by plugging into the formula $r = \frac{a_{i + 1}}{a_{i}}$ and simplifying.

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Step 2.1

Substitute $a_{i}$ and $a_{i + 1}$ into the formula for $r$ .

$r = \frac{5 \cdot {0.5}^{i + 1 - 1}}{5 \cdot {0.5}^{i - 1}}$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $5$ .

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Step 2.2.1.1

Cancel the common factor.

$r = \frac{5 \cdot {0.5}^{i + 1 - 1}}{5 \cdot {0.5}^{i - 1}}$

Step 2.2.1.2

Rewrite the expression.

$r = \frac{{0.5}^{i + 1 - 1}}{{0.5}^{i - 1}}$

Step 2.2.2

Cancel the common factor of ${0.5}^{i + 1 - 1}$ and ${0.5}^{i - 1}$ .

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Step 2.2.2.1

Factor ${0.5}^{i - 1}$ out of ${0.5}^{i + 1 - 1}$ .

$r = \frac{{0.5}^{i - 1} \cdot {0.5}^{i + 0 - (i - 1)}}{{0.5}^{i - 1}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

$r = \frac{{0.5}^{i - 1} \cdot {0.5}^{i + 0 - (i - 1)}}{{0.5}^{i - 1} \cdot 1}$

Step 2.2.2.2.2

Cancel the common factor.

$r = \frac{{0.5}^{i - 1} \cdot {0.5}^{i + 0 - (i - 1)}}{{0.5}^{i - 1} \cdot 1}$

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{{0.5}^{i + 0 - (i - 1)}}{1}$

Step 2.2.2.2.4

Divide ${0.5}^{i + 0 - (i - 1)}$ by $1$ .

$r = {0.5}^{i + 0 - (i - 1)}$

Step 2.2.3

Add $i$ and $0$ .

$r = {0.5}^{i - (i - 1)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Apply the distributive property.

$r = {0.5}^{i - i - - 1}$

Step 2.2.4.2

Multiply $- 1$ by $- 1$ .

$r = {0.5}^{i - i + 1}$

Step 2.2.5

Subtract $i$ from $i$ .

$r = {0.5}^{0 + 1}$

Step 2.2.6

Add $0$ and $1$ .

$r = {0.5}^{1}$

Step 2.2.7

Evaluate the exponent.

$r = 0.5$

Step 3

Since $| r | < 1$ , 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 $1$ for $i$ into $5 \cdot {0.5}^{i - 1}$ .

$a = 5 \cdot {0.5}^{1 - 1}$

Step 4.2

Simplify.

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Step 4.2.1

Subtract $1$ from $1$ .

$a = 5 \cdot {0.5}^{0}$

Step 4.2.2

Anything raised to $0$ is $1$ .

$a = 5 \cdot 1$

Step 4.2.3

Multiply $5$ by $1$ .

$a = 5$

Step 5

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

$\frac{5}{1 - 0.5}$

Step 6

Simplify.

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Step 6.1

Subtract $0.5$ from $1$ .

$\frac{5}{0.5}$

Step 6.2

Divide $5$ by $0.5$ .

$10$