Chapter 2: Problem 7
\(\sum_{n=0}^{\infty} \frac{(-1)^{n} z^{2 n}}{(2 n) !}\)
Short Answer
Expert verified
\( \cos(z) \)
Step by step solution
01
- Identify the series
Recognize that the given series \(\sum_{n=0}^{\infty} \frac{(-1)^{n} z^{2 n}}{(2 n) !}\) is a power series with the general term given by \(\frac{(-1)^{n} z^{2 n}}{(2 n)!}\).
02
- Compare with known series
Compare this series with the known Taylor series for common functions. Notice that this resembles the series expansion for the cosine function, \[ \cos(z) = \sum_{n=0}^{\infty} \frac{(-1)^{n} z^{2 n}}{(2 n) !}\].
03
- Write the result
Since the given series \( \sum_{n=0}^{\infty} \frac{(-1)^{n} z^{2 n}}{(2 n) !} \) matches the Taylor series for \( \cos(z) \), the sum of the series is \( \cos(z) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Taylor series
The Taylor series is an important concept in calculus and mathematical analysis. It is used to represent functions as infinite sums of terms calculated from the values of their derivatives at a single point. The general form of the Taylor series for a function \( f(x) \) around the point \( a \) is given by:
\[ f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots \]
Here are some important points to remember about the Taylor series:
\[ f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots \]
Here are some important points to remember about the Taylor series:
- Each term in the Taylor series involves the function's derivatives evaluated at a specific point, usually written as \( f^{(n)}(a) \).
- When the center of the series is at \( a = 0 \), it is referred to as the Maclaurin series. For example, the Taylor series of \( \text{cosine function} \) at \( a = 0 \) is a Maclaurin series.
- The Taylor series can approximate functions to a high degree of accuracy around the point \( a \).
cosine function
The cosine function (\text{cos}) is a fundamental trigonometric function. It is an even function, meaning \( \text{cos}(-x) = \text{cos}(x) \). It is widely used in oscillatory motions, wave behaviors, and many areas of mathematics and physics. Here is an essential property of the cosine function:
\[ \text{cos}(z) = \sum_{{n=0}}^{\infty} \frac{(-1)^{n} z^{2n}}{(2n)!} \]
This series expansion allows us to write \( \text{cos}(z) \) as an infinite sum of polynomial terms involving powers of \( z \).
- It is periodic with a period of \( 2\text{pi} \), meaning \ \( \text{cos}(x + 2\text{pi}) = \text{cos}(x) \).
- The range of the cosine function is from -1 to 1, inclusive.
- At \( x = 0, \text{cos}(0) = 1 \), and at \ \( x = \text{pi}/2, \text{cos}(\text{pi}/2) = 0 \).
\[ \text{cos}(z) = \sum_{{n=0}}^{\infty} \frac{(-1)^{n} z^{2n}}{(2n)!} \]
This series expansion allows us to write \( \text{cos}(z) \) as an infinite sum of polynomial terms involving powers of \( z \).
series expansion
Series expansion is a method for expressing a function as a sum of its terms, which can go up to infinity. Series expansions are particularly useful for approximating functions and solving differential equations. The most common types are the Taylor series and the Maclaurin series.
Some key points to remember about series expansion:
This representation breaks down the function into an infinite sum that converges to the exact value of \( \text{cos}(z) \). It highlights the elegance and power of mathematical series in describing functions.
Some key points to remember about series expansion:
- A series expansion allows us to approximate complex functions with a finite number of simpler terms.
- In the case of Taylor and Maclaurin series, these terms involve the function's derivatives at a specific point.
- The more terms we include in the series expansion, the more accurate the approximation becomes.
This representation breaks down the function into an infinite sum that converges to the exact value of \( \text{cos}(z) \). It highlights the elegance and power of mathematical series in describing functions.