First, let’s look at two simple nested loops first: for i (1..N) { for j (1..N) { f(); } } for i (1..N) { for j (i..N) { g(); } } f() is called N*N = N2 = O(N2) times. g() is called N+(N-1)+…+5+4+3+2+1 = N(N+1)/2 = N2/2 + N/2 = O(N2) times. As you … Read more
The number of iterations of the while-loop is exactly floor(x/y). Each iteration takes n operations, because that is the complexity of the subtraction r – y. Hence the complexity of the algorithm is n * floor(x/y). However, we want to express the complexity as a function of n, not as a function of x and … Read more
There will be exactly n iterations of the first loop. There will be exactly n(n+1)/2 iterations of the second loop. Inside of the second loop there is roughly 1/3 of n/2 long loops, 1/3 of j long loops and 1/3 of 2 long loops. If we look only at the first of the three possible … Read more
How about a_n = (-1)^n / sqrt(n) (source) For positive a_n there is no such sequence. Next time, you might want to ask at math.SE. solved sequence whose series is convergent, but square is not convergent [closed]