Time Complexity of Loop with Powers

What is the time complexity of below function?

void fun(int n, int k) 
{ 
    for (int i=1; i<=n; i++) 
    { 
      int p = pow(i, k);  
      for (int j=1; j<=p; j++) 
      { 
          // Some O(1) work 
      } 
    } 
} 

Time complexity of above function can be written as 1^k + 2^k + 3^k + … n1^k.

Let us try few examples:

k=1
Sum = 1 + 2 + 3 ... n 
    = n(n+1)/2 
    = n²/2 + n/2

k=2
Sum = 1² + 2² + 3² + ... n1².
    = n(n+1)(2n+1)/6
    = n³/3 + n²/2 + n/6

k=3
Sum = 1³ + 2³ + 3³ + ... n1³.
    = n²(n+1)²/4
    = n⁴/4 + n³/2 + n²/4

In general, asymptotic value can be written as (n^k+1)/(k+1) + Θ(n^k)

Note that, in asymptotic notations like Θ we can always ignore lower order terms. So the time complexity is Θ(n^k+1 / (k+1))

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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