To solve the given problem, we first need to understand the series in the denominator. The series given is:
[tex]\[ 1 + x + x^2 + x^3 + \ldots + x^{\infty} \][/tex]
This is an infinite geometric series with the first term [tex]\(a = 1\)[/tex] and the common ratio [tex]\(r = x\)[/tex].
For a geometric series with [tex]\(|r| < 1\)[/tex], the sum can be expressed as:
[tex]\[ \frac{a}{1 - r} \][/tex]
In our case, [tex]\(a = 1\)[/tex] and [tex]\(r = x\)[/tex]. Thus, the series sum is:
[tex]\[ \frac{1}{1 - x} \][/tex]
Therefore, the expression we need to maximize becomes:
[tex]\[ \frac{x^{100}}{\frac{1}{1 - x}} \][/tex]
Simplifying this, we get:
[tex]\[ \frac{x^{100} (1 - x)} \][/tex]
Next, we need to analyze the behavior of this new expression as [tex]\( x \)[/tex] approaches various values within its domain. Since [tex]\( x > 0 \)[/tex], let's consider the limit of this expression:
[tex]\[ \lim_{x \to 1} \frac{x^{100}}{\frac{1}{1 - x}} \][/tex]
Substituting [tex]\(x = 1\)[/tex] into the expression would make the denominator [tex]\((1 - 1) = 0\)[/tex], which gives us division by zero, indicating that the expression is no longer valid or defined at [tex]\( x = 1 \)[/tex]. This implies that as [tex]\( x \)[/tex] approaches 1 from the left, the fraction goes to [tex]\(\infty\)[/tex] causing the series sum to explode to [tex]\(\infty\)[/tex], and thus does not converge for [tex]\( x \geq 1 \)[/tex].
Now, we need to consider the limit as [tex]\(x \)[/tex] approaches [tex]\(0^+\)[/tex]:
[tex]\[ \lim_{x \to 0^+} \frac{x^{100}(1 - x)} \][/tex]
As [tex]\( x \to 0^+\)[/tex], [tex]\( x^{100} \to 0 \)[/tex] and [tex]\( (1 - x) \to 1\)[/tex]. Therefore, the expression approaches:
[tex]\[ 0 \][/tex]
So, as [tex]\(x \)[/tex] approaches [tex]\(0\)[/tex], the value of the expression approaches [tex]\(0\)[/tex].
Given the exploration of the boundary condition and the limit analysis, let's consider the value throughout the valid range [tex]\(0 < x < 1\)[/tex]. The expression [tex]\(x^{100}(1 - x)\)[/tex] is consistently positive decreasing towards zero, but we can conclude that the maximum value of the given expression across the defined domain is not infinite, it rather continues to go down to zero proving the leading result:
Therefore, the greatest value of the expression is:
[tex]\[0 \][/tex]
Which confirms all derived steps and analysis for the problem statement, reinforcing that maximum achievable value equals 0.
