How to find the fractional notation for the given infinite sum below?
#0.bar4# = #4/10# + #4/100# + #4/1000# + #4/10000# +.....

Question

Liam Brennan · Accepted Answer

The answer is #=4/9# Let #X=0.4444444444....# Then, #10X=4.4444444....# Therefore, #10X-X=4.4444444..-0.4444444....=4.00000# #9X=4# #X=4/9#

Liam Brennan · Answer

# = 4/9 # Notice how this is a geometric series with first term, #a = 4/10# and a common ratio of #1/10 # We know #S_oo = a/(1-r) # #= (4/10 )/ (1-1/10) # # = (4/10)/(9/10) # # = 4/9 #

Xavier Alonso · Answer

undefined To find the fractional notation for the given infinite sum $0.\overline{4} = \frac{4}{10} + \frac{4}{100} + \frac{4}{1000} + \frac{4}{10000} + \ldots$, we can express it as a geometric series.

Let $S$ be the sum of the series:

$$S = \frac{4}{10} + \frac{4}{100} + \frac{4}{1000} + \frac{4}{10000} + \ldots$$

Multiplying both sides of the equation by 10, we get:

$$10S = 4 + \frac{4}{10} + \frac{4}{100} + \frac{4}{1000} + \ldots$$

Subtracting the original equation from this, we get:

$$10S - S = 4$$

$$9S = 4$$

$$S = \frac{4}{9}$$

So, the fractional notation for the given infinite sum is $\frac{4}{9}$.

Elias Ackerman · Answer

undefined To find the fractional notation for the given infinite sum $ 0.\overline{4} = \frac{4}{10} + \frac{4}{100} + \frac{4}{1000} + \frac{4}{10000} + \ldots $, you can use the concept of geometric series.

First, let's express the infinite sum as a geometric series with common ratio $ \frac{1}{10} $, starting with the first term $ \frac{4}{10} $:
$$ S = \frac{4}{10} + \frac{4}{10^2} + \frac{4}{10^3} + \frac{4}{10^4} + \ldots $$

Now, we know that the sum of an infinite geometric series with a common ratio $ r $ and first term $ a $ is given by:
$$ S = \frac{a}{1 - r} $$

In our case, $ a = \frac{4}{10} $ and $ r = \frac{1}{10} $. Substituting these values into the formula, we get:
$$ S = \frac{\frac{4}{10}}{1 - \frac{1}{10}} $$

Simplifying the denominator:
$$ S = \frac{\frac{4}{10}}{\frac{9}{10}} $$
$$ S = \frac{4}{9} $$

Therefore, the fractional notation for the given infinite sum $ 0.\overline{4} $ is $ \frac{4}{9} $.

How to find the fractional notation for the given infinite sum below? #0.bar4# = #4/10# + #4/100# + #4/1000# + #4/10000# +.....