Question

Let \( f(x) = \sum_{n=0}^{\infty} (-1)^n x(x - 1)^n \) for \( 0 < x < 2 \). Then the value of \( f \left( \frac{\pi}{4} \right) \) is .............

Hint:

For evaluating power series, directly substitute the value of \( x \) and use standard results or computational tools to find the sum.

Answer 1

Correct Answer: 1

Step 1: Analyze the series.
We are given the series \( f(x) = \sum_{n=0}^{\infty} (-1)^n x (x - 1)^n \). This is a power series in terms of \( (x - 1) \). To evaluate at \( x = \frac{\pi}{4} \), we substitute \( x = \frac{\pi}{4} \) into the series.

Step 2: Evaluate the series.
We use the standard result for power series or directly evaluate the series at \( x = \frac{\pi}{4} \). After computation, we find that the value of the series at \( x = \frac{\pi}{4} \) is \( \boxed{1} \).

Step 3: Conclusion.
The value of \( f \left( \frac{\pi}{4} \right) \) is 1.