Calculus Examples

$\sum_{i = 0}^{\infty} {2.5}^{i}$

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.

Tap for more steps...

Step 2.1

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

$r = \frac{{2.5}^{i + 1}}{{2.5}^{i}}$

Step 2.2

Cancel the common factor of ${2.5}^{i + 1}$ and ${2.5}^{i}$ .

Tap for more steps...

Step 2.2.1

Factor ${2.5}^{i}$ out of ${2.5}^{i + 1}$ .

$r = \frac{{2.5}^{i} \cdot 2.5}{{2.5}^{i}}$

Step 2.2.2

Cancel the common factors.

Tap for more steps...

Step 2.2.2.1

Multiply by $1$ .

$r = \frac{{2.5}^{i} \cdot 2.5}{{2.5}^{i} \cdot 1}$

Step 2.2.2.2

Cancel the common factor.

$r = \frac{{2.5}^{i} \cdot 2.5}{{2.5}^{i} \cdot 1}$

Step 2.2.2.3

Rewrite the expression.

$r = \frac{2.5}{1}$

Step 2.2.2.4

Divide $2.5$ by $1$ .

$r = 2.5$

Step 3

Check if the series is convergent or divergent.

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