To find the value of x in the given equation, we need to analyze the given series and try to simplify it to match the form of the exponential function, e^x.

Given series: Σ(k=0 to n) nCk/n^k(k^3)

1. Simplifying the series:
Let's start by expanding the terms in the series:

= nC0/n^0(0^3) + nC1/n^1(1^3) + nC2/n^2(2^3) + ... + nCn/n^n(n^3)

= 1/1(0^3) + n/n(1^3) + n(n-1)/n^2(2^3) + ... + n!/(n^n)(n^3)

= 1/0 + 1 + (n-1)(n)/2^3 + ... + n!/(n^n)(n^3)

The term 1/0 is undefined, so we can ignore it. Also, notice that each term in the series involves the factor n^3. We can bring it out of the series as a constant factor:

= n^3(1 + n(n-1)/2^3 + ... + n!/(n^n))

2. Simplifying the inner series:
Now let's focus on simplifying the inner series:

1 + n(n-1)/2^3 + ... + n!/(n^n)

This inner series can be recognized as a partial sum of the Taylor series expansion for e^x, where x = n(n-1)/2^3 + ... + n!/(n^n).

3. Comparing with the Taylor series:
The Taylor series expansion for e^x is given by:

e^x = 1 + x + x^2/2! + x^3/3! + ...

Comparing the inner series with the Taylor series, we can see that they match if we let x = n(n-1)/2^3 + ... + n!/(n^n).

4. Finding the value of x:
Therefore, x = n(n-1)/2^3 + ... + n!/(n^n).

5. Conclusion:
The value of x in the given equation is n(n-1)/2^3 + ... + n!/(n^n).

X=2 ...