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Magazine Chapter 2: Problem 24
If \(f(x)=x^{n}\), then the value of \(f(1)-\frac{f^{\prime}(1)}{1 !}+\frac{f^{\prime \prime}(1)}{2 !}-\frac{f^{\prime \prime \prime}(1)}{3 !}+\frac{f^{(\omega)}(1)}{4 !}-\ldots\) \(\ldots+\frac{(-1)^{n} f^{n}(1)}{n !}\) is (a) 1 (b) \(2^{n}\) (c) \(2^{n-1}\) (d) 0
Short Answer
Expert verified
The value of the expression is 0.
Step by step solution
01
Understand the Problem
The problem involves computing a specific sum where each term relates to the derivative of the function \(f(x) = x^n\) at \(x = 1\). We need to calculate \(f(1)\), then each successive derivative evaluated at 1, and apply the alternating sum formula.
02
Calculate the Function Value at 1
First, substitute \(x = 1\) into the function: \(f(1) = 1^n = 1\). This is because any number raised to the power of \(n\) will still be 1 when that number is 1.
03
Derive and Evaluate First Derivative at 1
The first derivative of \(f(x) = x^n\) is given by \(f'(x) = nx^{n-1}\). Evaluating at \(x = 1\) gives \(f'(1) = n \times 1^{n-1} = n\).
04
Derive and Evaluate Second Derivative at 1
The second derivative is \(f''(x) = n(n-1)x^{n-2}\). Evaluating at 1, \(f''(1) = n(n-1) \times 1^{n-2} = n(n-1)\).
05
Derive Higher-Order Derivatives at 1
Continuing in this pattern, for the \(k\)-th derivative evaluated at 1, we have \(f^{(k)}(1) = n(n-1)(n-2) \, ... \, (n-k+1)\), provided \(k \leq n\). For \(k > n\), all terms are zero because all the differentiations will result in zero factors.
06
Set Up the Alternating Sum
The expression is \(f(1) - \frac{f'(1)}{1!} + \frac{f''(1)}{2!} - \frac{f'''(1)}{3!} + \ldots + \frac{(-1)^n f^{(n)}(1)}{n!}\). This sum alternates signs.
07
Simplify Using the Binomial Expansion
Notice that this sum represents the expansion of \((1-1)^n\) using Taylor's formula, which evaluates to zero when tested for any positive integer \(n\), indicating that all terms cancel out appropriately.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Taylor Expansion
Taylor Expansion is a method used to express a function as an infinite sum of terms. Each term is calculated from the derivatives of the function at a single point. This expression helps approximate complex functions using polynomials.
In essence, a Taylor series is centered around a specific point, often denoted by "a." The function value and its derivatives evaluated at this specific point form the building blocks of the series.
For a function, the Taylor series around the point "a" is given by:
In essence, a Taylor series is centered around a specific point, often denoted by "a." The function value and its derivatives evaluated at this specific point form the building blocks of the series.
For a function, the Taylor series around the point "a" is given by:
- First, the value of the function at "a": \( f(a) \).
- Subsequently, derivatives like \( f'(a) \), \( f''(a) \), etc., contribute each term.
- The general term looks like: \( \frac{f^{(n)}(a)}{n!} (x-a)^n \).
Higher-Order Derivatives
Higher-order derivatives extend the concept of a derivative beyond the first order. The first derivative gives the rate of change of a function, but this can be taken further.
The subsequent derivatives provide more detailed information about the function's behavior. Here's how they work:
The subsequent derivatives provide more detailed information about the function's behavior. Here's how they work:
- First derivative, \( f'(x) \), represents the slope or rate of change.
- Second derivative, \( f''(x) \), indicates how the slope itself is changing, often related to the function's concavity.
- Third and higher-order derivatives reveal even subtler changes.
Alternating Series
An alternating series is a sequence of numbers where the signs of the terms alternate. Often expressed as +, -, +, -, alternating series converge differently and can simplify to specific results.
In our context, the problem relies on an alternating sum derived from function derivatives evaluated at one point: \( f(1) - \frac{f'}{1!} + \frac{f''}{2!} - \ldots \).
Such series exhibit characteristics:
In our context, the problem relies on an alternating sum derived from function derivatives evaluated at one point: \( f(1) - \frac{f'}{1!} + \frac{f''}{2!} - \ldots \).
Such series exhibit characteristics:
- They can simplify quickly due to terms canceling each other.
- Convergence can be ensured under conditions such as decreasing absolute values of succeeding terms.
- Useful for simplifying expressions, especially in problems like ours, which involve evaluating Taylor expansions over consistent subset sequences.
Binomial Theorem
The Binomial Theorem is a formula for expanding expressions that are raised to any power, such as \((a+b)^n\). It establishes a clear way to compute or approximate such powers.
Here鈥檚 a simple breakdown of its application:
Such cancellations mirror outcomes from our alternating series, showcasing a zero result as expected from both methods. Hence, the Binomial Theorem delivers not only insights into polynomial expansions but also aligns with alternating sequences in complex expressions.
Here鈥檚 a simple breakdown of its application:
- Each term in the expansion involves a binomial coefficient, \( \binom{n}{k} \), which relates to combinations of elements.
- The terms are of the form \( a^{n-k}b^k \).
Such cancellations mirror outcomes from our alternating series, showcasing a zero result as expected from both methods. Hence, the Binomial Theorem delivers not only insights into polynomial expansions but also aligns with alternating sequences in complex expressions.
