Find the value of 1/10+1/100+1/1000+1/10000+...
Leave your answer as a fraction
you have a geometric series
with
a = 1/10 and r = 1/10
sum(∞ = a/(1-r)
= (1/10) / (9/10)
= (1/10)(10/9) = 1/9
To find the value of the infinite series 1/10 + 1/100 + 1/1000 + 1/10000 + ..., we can use the concept of geometric progression.
A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant, called the common ratio (r).
In this case, our sequence has a first term of 1/10 and a common ratio of 1/10.
The sum of an infinite geometric series can be calculated using the formula S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.
Plugging in the values for our series, we have a = 1/10 and r = 1/10. Thus, the sum of the series is:
S = (1/10) / (1 - 1/10)
To simplify this expression, we need to find a common denominator and then perform the division:
S = (1/10) / (9/10)
To divide by a fraction, we multiply by its reciprocal:
S = (1/10) * (10/9)
Multiplying across the numerators and denominators, we get:
S = 1/9
Therefore, the value of the infinite series 1/10 + 1/100 + 1/1000 + 1/10000 + ... is 1/9.