Question

What is the coefficient of $ x^{10} $ in the expansion of the generating function $$ \left( 1 + x + x^2 + x^3 + \cdots \right)^2? $$

• A. 1
• B. 10
• C. 2
• D. 11

Hint:

When dealing with generating functions, remember to first identify the series and then apply known series expansions to find the required coefficient.

Answer 1

The Correct Option is D

The generating function given is: \[ \left( 1 + x + x^2 + x^3 + \cdots \right)^2. \] This can be recognized as the square of a geometric series. The series \( 1 + x + x^2 + x^3 + \cdots \) is a geometric series with the first term \( 1 \) and the common ratio \( x \), so we can express it as: \[ \frac{1}{1-x}. \]
Thus, the given generating function becomes: \[ \left( \frac{1}{1 - x} \right)^2 = \frac{1}{(1 - x)^2}. \]
Step 1: Expanding \( \frac{1}{(1 - x)^2} \)
The series expansion of \( \frac{1}{(1 - x)^2} \) is given by the known result: \[ \frac{1}{(1 - x)^2} = \sum_{n=1}^{\infty} n x^{n-1}. \] This represents the expansion of \( \frac{1}{(1 - x)^2} \).
Step 2: Finding the Coefficient of \( x^{10} \)
To find the coefficient of \( x^{10} \), we need the term that corresponds to \( x^{10} \) in the expansion. The term for \( x^{10} \) comes from \( n = 11 \), because the general term is \( n x^{n-1} \).
Thus, the coefficient of \( x^{10} \) is \( 11 \).
Step 3: Conclusion
Therefore, the coefficient of \( x^{10} \) in the expansion of the generating function is \( 11 \).