Calculus Examples

$\sum_{i = 1}^{\infty} 61 \cdot {0.01}^{i}$

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.

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Step 2.1

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

$r = \frac{61 \cdot {0.01}^{i + 1}}{61 \cdot {0.01}^{i}}$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $61$ .

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Step 2.2.1.1

Cancel the common factor.

$r = \frac{61 \cdot {0.01}^{i + 1}}{61 \cdot {0.01}^{i}}$

Step 2.2.1.2

Rewrite the expression.

$r = \frac{{0.01}^{i + 1}}{{0.01}^{i}}$

Step 2.2.2

Cancel the common factor of ${0.01}^{i + 1}$ and ${0.01}^{i}$ .

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Step 2.2.2.1

Factor ${0.01}^{i}$ out of ${0.01}^{i + 1}$ .

$r = \frac{{0.01}^{i} \cdot 0.01}{{0.01}^{i}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

$r = \frac{{0.01}^{i} \cdot 0.01}{{0.01}^{i} \cdot 1}$

Step 2.2.2.2.2

Cancel the common factor.

$r = \frac{{0.01}^{i} \cdot 0.01}{{0.01}^{i} \cdot 1}$

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{0.01}{1}$

Step 2.2.2.2.4

Divide $0.01$ by $1$ .

$r = 0.01$

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 $61 \cdot {0.01}^{i}$ .

$a = 61 \cdot {0.01}^{1}$

Step 4.2

Simplify.

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Step 4.2.1

Evaluate the exponent.

$a = 61 \cdot 0.01$

Step 4.2.2

Multiply $61$ by $0.01$ .

$a = 0.61$

Step 5

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

$\frac{0.61}{1 - 0.01}$

Step 6

Simplify.

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Step 6.1

Subtract $0.01$ from $1$ .

$\frac{0.61}{0.99}$

Step 6.2

Divide $0.61$ by $0.99$ .

$0.\overline{61}$