Dashed lines in ACF plot in R

Question

I'm going through the book 'Introductory Time Series with R' by Cowpertwait and Metcalfe. On page 36 Its says the lines are at: $-1/n \pm 2/\sqrt{n}$. I've read here R forum that the lines are at $\pm 1.96/\sqrt{n}$.

I ran the following code :

b = c(3,1,4,1)

acf(b)

and I see that the lines look to appear to be at $\pm 1.96/\sqrt{4}$. So, obviously the book is wrong? Or, Am I misreading what has been written? Are the authors talking about something slightly different?

*Note, I'm not interested in the 1.96 vs 2 minor detail discrepancy. I assume this was just the author using the rule of thumb of 2 sd's versus the actual 1.96 sd.

Edit: I ran this simulation:

acf1 = 0
acf2 = 0
acf3 = 0
for(i in 1:5000){
  resids= runif(1000)
  residsacf = c(acf(resids,plot= FALSE))
  acf1[i] = residsacf$acf[2,,1]
  acf2[i] = residsacf$acf[3,,1]
  acf3[i] = residsacf$acf[4,,1]
}
meanacf1 = mean(acf1)
meanacf2 = mean(acf2)
meanacf3 = mean(acf3)
meanacf1
meanacf2
meanacf3

I always seem to get values near $1/n$ for all 3.

Further edit : I'm seeing a trend of $1/n-(k-1)/n^2$

Answer 1

The sample autocorrelation is negatively biased and the first sample autocorrelation coefficient has mean $-1/n$ where $n$ is the number of observations. But Metcalfe and Cowpertwait are incorrect in saying that all autocorrelation coefficients have that mean, and they are also incorrect in saying that R plots lines at $-1/n \pm 1.96/\sqrt{n}$.

Asymptotically the mean is 0 and that is what R uses in plotting the lines at $\pm 1.96/\sqrt{n}$ .