This is a problem on sequences and series involving describing a repeating decimal as an infinite series in order to express it as a ratio of two integers; for some reason, I'm off by a factor of ten, and I can't figure why. Posted below in a minute.

Question

mendicant_bias · Answer

The repeating decimal is just 0.777-

$$0.777... = \frac{ 7 }{ 10 }+\frac{ 7 }{ 100 }+ \frac{ 7 }{ 1000 } ... \sum_{n = 1}^{\infty}\frac{ 7 }{ 10^{n} }$$

campbell_st · Answer

ok... so the 1st term is 7/10 and the common ratio is 1/10

campbell_st · Answer

the series is geometric and converges... since  -1 < r < 1

mendicant_bias · Answer

Since it's obviously a geometric series, 

$$\sum_{n = 1}^{\infty}\frac{ 7 }{ 10^{n} }, a = 7, r = \frac{ 1 }{ 10 }, ar ^{n-1}$$
The infinite sum for a geometric series is $$\frac{ a }{ 1-r }$$
$$\frac{ 7 }{ 1-(\frac{ 1 }{ 10 }) } = \frac{ 7 }{ \frac{ 9 }{ 10 } } = \frac{ 70 }{ 9 }$$

mendicant_bias · Answer

The answer is 7/9ths. I can't figure out why I'm off by a factor of ten.  But yeah, full aware of the common ratio, and convergence is a given. I'm just looking for a simple algebraic error or something.

campbell_st · Answer

so the way to express it as an rational number is 

let x = 0.7777..... 

multiply it by 10

so 10x = 7.777777777

so now subtract the 2nd equation from the 1st

    10x =  7.7777777777777....
-      x = 0.7777777777777.....
--------------------------

9x = 7

campbell_st · Answer

so now solve for x

mendicant_bias · Answer

I can do that, too, but I'd like to use the formula, rather than just plug and chug. I know how to do it otherwise, but I'm trying to do it with the formula.

campbell_st · Answer

well its not plug and chug, its a valid method... 

so what you need to do is find the limiting sum of the geometric series

$$S_{\infty} = \frac{a}{1 - r}$$

campbell_st · Answer

thats if you want to solve it as a series

mendicant_bias · Answer

Yeah, got that. I know it's a-that's exactly what I'm trying to do, and why I wrote the formula.

campbell_st · Answer

so you have 

$$S_{\infty} = \frac{\frac{7}{10}}{1 - \frac{1}{10}}$$

mendicant_bias · Answer

a is now 7/10.

mendicant_bias · Answer

Thanks, that's what I was looking for; It was just the number. I understand the process completely, but I couldn't figure out what a should be. That's what I needed.

campbell_st · Answer

a is  7/10  1st decimal place

mendicant_bias · Answer

Got it.