Calculus Examples

$\sum_{i = 1}^{\infty} \frac{4}{7} {(\frac{7}{6})}^{i - 1}$

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 $r$ .

$r = \frac{\frac{4}{7} \cdot {(\frac{7}{6})}^{i + 1 - 1}}{\frac{4}{7} \cdot {(\frac{7}{6})}^{i - 1}}$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $\frac{4}{7}$ .

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Step 2.2.1.1

Cancel the common factor.

$r = \frac{\frac{4}{7} \cdot {(\frac{7}{6})}^{i + 1 - 1}}{\frac{4}{7} \cdot {(\frac{7}{6})}^{i - 1}}$

Step 2.2.1.2

Rewrite the expression.

$r = \frac{{(\frac{7}{6})}^{i + 1 - 1}}{{(\frac{7}{6})}^{i - 1}}$

Step 2.2.2

Cancel the common factor of ${(\frac{7}{6})}^{i + 1 - 1}$ and ${(\frac{7}{6})}^{i - 1}$ .

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Step 2.2.2.1

Factor ${(\frac{7}{6})}^{i - 1}$ out of ${(\frac{7}{6})}^{i + 1 - 1}$ .

$r = \frac{{(\frac{7}{6})}^{i - 1} {(\frac{7}{6})}^{i + 0 - (i - 1)}}{{(\frac{7}{6})}^{i - 1}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

$r = \frac{{(\frac{7}{6})}^{i - 1} {(\frac{7}{6})}^{i + 0 - (i - 1)}}{{(\frac{7}{6})}^{i - 1} \cdot 1}$

Step 2.2.2.2.2

Cancel the common factor.

$r = \frac{{(\frac{7}{6})}^{i - 1} {(\frac{7}{6})}^{i + 0 - (i - 1)}}{{(\frac{7}{6})}^{i - 1} \cdot 1}$

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{{(\frac{7}{6})}^{i + 0 - (i - 1)}}{1}$

Step 2.2.2.2.4

Divide ${(\frac{7}{6})}^{i + 0 - (i - 1)}$ by $1$ .

$r = {(\frac{7}{6})}^{i + 0 - (i - 1)}$

Step 2.2.3

Add $i$ and $0$ .

$r = {(\frac{7}{6})}^{i - (i - 1)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Apply the distributive property.

$r = {(\frac{7}{6})}^{i - i - - 1}$

Step 2.2.4.2

Multiply $- 1$ by $- 1$ .

$r = {(\frac{7}{6})}^{i - i + 1}$

Step 2.2.5

Subtract $i$ from $i$ .

$r = {(\frac{7}{6})}^{0 + 1}$

Step 2.2.6

Add $0$ and $1$ .

$r = {(\frac{7}{6})}^{1}$

Step 2.2.7

Simplify.

$r = \frac{7}{6}$

Step 3

Check if the series is convergent or divergent.

Since $| r | \geq 1$ , the series diverges.