Time Complexity of Loop with Powers

What is the time complexity of the below function?

C++

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
        }
    }
}

// This code is contributed by Shubham Singh

C

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
        }
    }
}

Java

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

// This code is contributed by umadevi9616

Python3

def fun(n, k):

    for i in range(1, n + 1):
        p = pow(i, k)
        for j in range(1, p + 1):
            # Some O(1) work


# This code is contributed by Shubham Singh

C#

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

// This code is contributed by umadevi9616

Javascript

<script>

// JavaScript program for the above approach
function fun(n, k)
{
    for(let i = 1; i <= n; i++)
    {
        int p = Math.pow(i, k);
        for (let j = 1; j <= p; j++)
        {
            // Some O(1) work
        }
    }
}

// This code is contributed by Shubham Singh

</script>

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)
If  n>=k then the time complexity will be considered in O((n^k+1)/(k+1)) and if n<k, then the time complexity will be considered as  in the O(n^k)

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