Binomial Expansion Formula

Now,


Hence, it is divisible by .

We are tasked with finding the term independent of in the expansion of: This involves using the binomial expansion and focusing on the term that is independent of .
Step 1: Approximate for large .
Since for large , we approximate: Thus, the expression becomes:
Step 2: Binomial Expansion.
The binomial expansion of is given by: Simplifying each term: This simplifies to:
Step 3: Finding the Independent Term.
For the term to be independent of , the exponent of must be zero: Since must be an integer, must be a multiple of 5 for the term independent of to exist.
Step 4: Asymptotic Behavior.
As becomes very large, the coefficient of the term independent of behaves in such a way that: This is because the factorial growth of is not sufficient to offset the exponential decay due to the powers of and in the denominator.
Step 5: Conclusion.
The value of the limit is .