Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

If f(x)=∑ₖ₌₀∞ cₖ xᵏ=0, for all x on an interval (−a, a), then cₖ = 0, for all k.

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Recognize that the function \( f(x) = \sum_{k=0}^{\infty} c_k x^k \) is represented as a power series centered at 0, which converges on the interval \((-a, a)\).

Recall the uniqueness theorem for power series: if a power series converges to zero for all \( x \) in an interval around zero, then all its coefficients must be zero.

Understand that this theorem follows from the fact that power series are analytic functions, and an analytic function that is identically zero on an interval must have all zero coefficients in its power series expansion.

To see why, consider differentiating \( f(x) \) term-by-term and evaluating at \( x=0 \). The \( k \)-th derivative at zero is related to \( c_k \) by \( f^{(k)}(0) = k! c_k \). Since \( f(x) = 0 \) for all \( x \), all derivatives at zero must be zero, implying \( c_k = 0 \) for all \( k \).

Therefore, the statement is true: if \( f(x) = 0 \) for all \( x \) in \((-a, a)\), then all coefficients \( c_k \) must be zero.

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Power Series and Radius of Convergence

A power series is an infinite sum of the form ∑ cₖ xᵏ, which converges within a certain radius around zero called the radius of convergence. Within this interval, the series represents a function that can be analyzed term-by-term. Understanding convergence is essential to determine where the series equals zero.

Uniqueness Theorem for Power Series

If a power series converges to zero for all x in an interval around zero, then all its coefficients must be zero. This theorem ensures that the zero function has a unique power series representation, meaning no nontrivial power series can vanish identically on an interval without all coefficients being zero.

Counterexamples and Zero Functions

To verify the statement, one must consider whether a nonzero power series can equal zero on an interval. Counterexamples involve functions that vanish at points but not on an entire interval. The concept clarifies that if the function is zero on an interval, the coefficients must be zero, reinforcing the uniqueness of power series expansions.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.If f(x)=∑ₖ₌₀∞ cₖ xᵏ=0, for all x on an interval (−a, a), then cₖ = 0, for all k.