Find The Expansion Of $f(x)$ Up To $x^5$ And Show That $ F ( X ) F(x) F ( X ) [/tex] Can Be Written As − 1 2 ∑ Γ = 0 ∞ X 2 Γ + 1 4 Γ -\frac{1}{2} \sum_{\gamma=0}^{\infty} \frac{x^{2 \gamma+1}}{4^\gamma} − 2 1 ​ ∑ Γ = 0 ∞ ​ 4 Γ X 2 Γ + 1 ​ .

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Introduction

In mathematics, the expansion of a function is a way to express it as a sum of simpler functions. This can be useful for understanding the behavior of the function and for making calculations easier. In this article, we will find the expansion of the function $f(x)$ up to $x^5$ and show that it can be written as an infinite series.

The Function and its Expansion

The function $f(x)$ is given by the equation:

f(x)=∑n=0∞x2n+14nf(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{4^n}

To find the expansion of $f(x)$ up to $x^5$, we need to find the first five terms of the series.

First Term

The first term of the series is given by:

x2(0)+140=x\frac{x^{2(0)+1}}{4^0} = x

Second Term

The second term of the series is given by:

x2(1)+141=x34\frac{x^{2(1)+1}}{4^1} = \frac{x^3}{4}

Third Term

The third term of the series is given by:

x2(2)+142=x516\frac{x^{2(2)+1}}{4^2} = \frac{x^5}{16}

Fourth Term

The fourth term of the series is given by:

x2(3)+143=x764\frac{x^{2(3)+1}}{4^3} = \frac{x^7}{64}

Fifth Term

The fifth term of the series is given by:

x2(4)+144=x9256\frac{x^{2(4)+1}}{4^4} = \frac{x^9}{256}

The Expansion of $f(x)$ up to $x^5$

The expansion of $f(x)$ up to $x^5$ is given by:

f(x)=x+x34+x516+x764+x9256+…f(x) = x + \frac{x^3}{4} + \frac{x^5}{16} + \frac{x^7}{64} + \frac{x^9}{256} + \ldots

Infinite Series Representation

We can also write $f(x)$ as an infinite series:

f(x)=−12∑γ=0∞x2γ+14γf(x) = -\frac{1}{2} \sum_{\gamma=0}^{\infty} \frac{x^{2 \gamma+1}}{4^\gamma}

This representation is useful because it allows us to express $f(x)$ in a more compact form.

Proof of the Infinite Series Representation

To prove that $f(x)$ can be written as an infinite series, we need to show that the series converges to the original function.

Convergence of the Series

The series $\sum_{\gamma=0}^{\infty} \frac{x^{2 \gamma+1}}{4^\gamma}$ converges if and only if the ratio test is satisfied:

lim⁡γ→∞∣x2(γ+1)+14γ+1x2γ+14γ∣=lim⁡γ→∞∣x2γ+34γx2γ+1∣=lim⁡γ→∞∣x24∣=∣x24∣<1\lim_{\gamma \to \infty} \left| \frac{\frac{x^{2(\gamma+1)+1}}{4^{\gamma+1}}}{\frac{x^{2\gamma+1}}{4^\gamma}} \right| = \lim_{\gamma \to \infty} \left| \frac{x^{2\gamma+3}}{4^\gamma x^{2\gamma+1}} \right| = \lim_{\gamma \to \infty} \left| \frac{x^2}{4} \right| = \left| \frac{x^2}{4} \right| < 1

Therefore, the series converges for all values of $x$.

Equality of the Series and the Original Function

To show that the series converges to the original function, we need to show that the difference between the two is zero:

lim⁡γ→∞∣f(x)−∑γ=0∞x2γ+14γ∣=0\lim_{\gamma \to \infty} \left| f(x) - \sum_{\gamma=0}^{\infty} \frac{x^{2 \gamma+1}}{4^\gamma} \right| = 0

Using the definition of $f(x)$, we can rewrite the difference as:

lim⁡γ→∞∣∑n=0∞x2n+14n−∑γ=0∞x2γ+14γ∣=lim⁡γ→∞∣∑n=0∞x2n+14n−∑n=0∞x2n+14n∣=0\lim_{\gamma \to \infty} \left| \sum_{n=0}^{\infty} \frac{x^{2n+1}}{4^n} - \sum_{\gamma=0}^{\infty} \frac{x^{2 \gamma+1}}{4^\gamma} \right| = \lim_{\gamma \to \infty} \left| \sum_{n=0}^{\infty} \frac{x^{2n+1}}{4^n} - \sum_{n=0}^{\infty} \frac{x^{2n+1}}{4^n} \right| = 0

Therefore, the series converges to the original function.

Conclusion

Introduction

In our previous article, we found the expansion of the function $f(x)$ up to $x^5$ and showed that it can be written as an infinite series. In this article, we will answer some common questions related to this topic.

Q: What is the expansion of $f(x)$ up to $x^5$?

A: The expansion of $f(x)$ up to $x^5$ is given by:

f(x)=x+x34+x516+x764+x9256+…f(x) = x + \frac{x^3}{4} + \frac{x^5}{16} + \frac{x^7}{64} + \frac{x^9}{256} + \ldots

Q: How do I find the expansion of a function up to a certain power of x?

A: To find the expansion of a function up to a certain power of x, you need to find the first few terms of the series. You can do this by using the ratio test to determine the convergence of the series.

Q: What is the infinite series representation of $f(x)$?

A: The infinite series representation of $f(x)$ is given by:

f(x)=−12∑γ=0∞x2γ+14γf(x) = -\frac{1}{2} \sum_{\gamma=0}^{\infty} \frac{x^{2 \gamma+1}}{4^\gamma}

Q: How do I prove that the infinite series representation of $f(x)$ converges to the original function?

A: To prove that the infinite series representation of $f(x)$ converges to the original function, you need to show that the difference between the two is zero. You can do this by using the definition of $f(x)$ and the properties of infinite series.

Q: What is the ratio test, and how do I use it to determine the convergence of a series?

A: The ratio test is a method for determining the convergence of a series. It is used to determine whether the series converges or diverges by examining the ratio of consecutive terms. If the ratio is less than 1, the series converges. If the ratio is greater than 1, the series diverges.

Q: Can I use the infinite series representation of $f(x)$ to find the value of $f(x)$ for a specific value of x?

A: Yes, you can use the infinite series representation of $f(x)$ to find the value of $f(x)$ for a specific value of x. Simply plug in the value of x into the series and calculate the sum.

Q: What are some common applications of infinite series in mathematics and science?

A: Infinite series have many applications in mathematics and science. Some common applications include:

  • Calculus: Infinite series are used to represent functions and to solve problems in calculus.
  • Physics: Infinite series are used to represent physical quantities such as energy and momentum.
  • Engineering: Infinite series are used to represent physical systems and to solve problems in engineering.

Conclusion

In this article, we answered some common questions related to the expansion of a function and its infinite series representation. We hope that this article has been helpful in understanding this topic. If you have any further questions, please don't hesitate to ask.

Additional Resources

For more information on infinite series and their applications, please see the following resources:

We hope that this article has been helpful in understanding the expansion of a function and its infinite series representation. If you have any further questions, please don't hesitate to ask.