Double summation without loops

[AlberteOlesen]

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

(as seen on this picture Double summation

I am quite lost with this exercise, I tried the following code but I it returns an error although giving me a number - pretty sure it is not correct. Any help is greatly valued!

  i <- seq(1, 10, 1) 
  j <- seq(1, i, 1)
  denominators <- 10+j^i
  fractions <- (i^5)/denominators
  sum(fractions) 

or

  i = rep(1:10, each=5)
  j = rep(i, 10)
  # find the sum 
  sum(i^5/(10+j^i))

[woodward]

One way you can do it by making the complete square of i, j and then ignoring the ones where j > i:

# make vectors of indices
i <- rep(1:10, each = 10)
j <- rep(1:10, times = 10)

# which ones are j <= i
k <- j <= i

# find the sum
sum(i[k]^5/(10+j[k]^i[k]))
#> [1] 20835.22

^{Created on 2020-11-13 by the reprex package (v0.3.0)}