Question

Use the formula to evaluate the infinite series. Round your answer to the nearest hundredth if necessary. 72 + 12 + 2 + . . .

Answer 1

To find the sum of an infinite series, use this following formula
S∞ = $\dfrac{a_{1}}{1-r}$
with S∞ as the sum of infinite series, a₁ as the first term of the series, and r as the ratio.

First, find the ratio of the series
The ratio can be defined using division of the nth term by the (n-1)th term. Or if you use a₂, the divisor will be a₍₂₋₁₎ = a₁
r = a₂/a₁
r = $\dfrac{12}{72}$
r = $\dfrac{1}{6}$
The ratio of the series is 1/6

Second, calculate the sum using the formula above.
S∞ = $\dfrac{a}{1-r}$

Plug in the numbers
S∞ = $\dfrac{72}{1- \frac{1}{6} }$

Write the division horizontally
S∞ = 72 ÷ $(1-\frac{1}{6})$

1 can be written as 6/6
S∞ = 72 ÷ $(\frac{6}{6} -\frac{1}{6} )$
S∞ = 72 ÷ $\frac{5}{6}$

Change the division into multiplication
S∞ = 72 × $\frac{6}{5}$
S∞ = $\frac{432}{5}$
S∞ = 86.4
The sum of the series is 86.4