# Summation notation index n-1 refers to non-existent element? [closed]

I don't have any math background, and I'm trying to type this equation into R, and having difficulties understanding the summation notation.

I dont understand the part, because it seems to refer to values which do not exist.

If the vector R for example has 3 elements: c( element_1 = 1, element_2 = 2, element_3 = 3 ), then it seems to me that the summation from j = 1 to to i-1 (for i=1 ) refers to indices from 1 to (1-1), that is, from 1 to 0. But the index zero does not exist in the vektor R.

So what is the the sigma referring to?

Here is what I typed into R sofar, it's everything but the

# Input data:
M <-   c(140L, 12L, 12L, 59L, 94L, 101L, 117L, 213L, 368L, 607L, 1025L,
1488L, 2255L, 2787L, 3257L, 3715L, 4231L, 6281L)

R <-  c(42L, 22L, 28L, 54L, 77L, 108L, 169L, 227L, 293L, 531L, 863L,
1464L, 2591L, 3334L, 3045L, 2605L, 1890L, 1261L)

D <-  c(2L, 1L, 2L, 6L, 4L, 7L, 15L, 26L, 67L, 120L, 304L, 497L, 883L,
1158L, 1321L, 1318L, 1177L, 1065L)

N <- c(167323L, 168088L, 176017L, 180986L, 168189L, 155506L, 174274L,
195538L, 207287L, 204711L, 183802L, 174342L, 183415L, 151277L,
104199L, 71782L, 47503L, 33946L)

# W width of age interval
w <-  c( 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,Inf )

# function
v1 <- numeric()

for(i in 1:length(R))  {

v1[i] <- R[i] /  ( R[i] + M[i] - D[i] )  *  ( 1 - exp( - (w[i]/N[i]) *  (R[i] + M[i] - D[i]) ) )

}

sum(v1)


## closed as off-topic by Alex M., Math1000, colormegone, Frits Veerman, egregApr 15 '16 at 22:54

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Alex M., Math1000
If this question can be reworded to fit the rules in the help center, please edit the question.

• I think this is really a question about the R programming language. I think that in R vectors are indexed starting with 0, so the first element is at position 0, not position 1. See if this helps. – Ethan Bolker Apr 15 '16 at 13:40
• Thanks. This a programming question, but, I think, also a math question, as G Cab's math-answer below helped me understand the programming question :) – Rasmus Larsen Apr 17 '16 at 10:57

1) $\quad \sum\limits_{j = 1}^{i - 1} {a_{\,j} } = \sum\limits_{1\, \le \,j\, \le \,i - 1} {a_{\,j} } \quad \Rightarrow \quad 0\quad \left| {\;i \le 1} \right.$
2) (antiderivative)$\quad a_{\,j} = b_{\,j + 1} - b_{\,j} \quad \Rightarrow \quad \sum\limits_{j = 1}^{i - 1} {a_{\,j} } = \sum\nolimits_{j = 1}^{\,i} {a_{\,j} } = b_{\,i} - b_{\,1} = - \sum\nolimits_{j = i}^{\,1} {a_{\,j} }$
For $i=1$ both acceptions give $0$ as a result.