Summation without loops

Question

I have following double summation: ∑10,i=1 ∑i,j=1 (i^5/(10+j^i))

I am quite lost with this exercise, I tried the following code but I it returns an error although giving

Answer 1

Make a function of the inner sum:

f <- Vectorize(function(i) {
         j <- 1:i
         sum(i^5 / (10 + j^i))
     })

By vectorizing it you can apply it to arrays, where it will operate component by component: that's what the outer sum over i says to do. Thus, the value is

sum(f(1:10))

---

Another solution, wasteful of RAM and a bit slower, exploits the outer product to compute all the terms of the double sum in one matrix. You have to extract the terms for which j does not exceed i:

n <- 10
x <- outer(1:n, 1:n, function(i,j) i^5 / (10 + j^i))
sum(x[!upper.tri(x)])

For its compactness and simplicity, though, it's a good technique to know.