1/x^p

Definition

The term 1/x^p, where p is a positive real number, represents a specific type of function that is commonly encountered in the context of the Divergence Test and the Integral Test in calculus. This function belongs to the class of power functions and exhibits unique properties that are crucial for understanding the convergence or divergence of series.

5 Must Know Facts For Your Next Test

1. The term $1/x^p$ is a decreasing function for $x > 0$ and $p > 0$, meaning it approaches 0 as $x$ approaches infinity.
2. The behavior of the series $ extbackslash sum_{n=1}^{ extbackslash infty} 1/n^p$ is determined by the value of $p$. If $p > 1$, the series converges, and if $p \leq 1$, the series diverges.
3. The Divergence Test states that if the general term of a series, $a_n$, satisfies $ extbackslash lim_{n \to extbackslash infty} a_n \neq 0$, then the series diverges.
4. The Integral Test can be used to determine the convergence or divergence of a series involving the term $1/x^p$ by comparing the series to the corresponding improper integral.
5. The behavior of the series $ extbackslash sum_{n=1}^{ extbackslash infty} 1/n^p$ is closely related to the convergence or divergence of the improper integral $ extbackslash int_1^{ extbackslash infty} 1/x^p \textbackslash, dx$.

Review Questions

• Explain how the term $1/x^p$ is related to the Divergence Test.
  • The term $1/x^p$ is a crucial component in the application of the Divergence Test. Since the Divergence Test states that a series diverges if the general term does not approach 0 as $n$ approaches infinity, the behavior of $1/x^p$ is particularly relevant. For $p > 1$, the term $1/x^p$ approaches 0 as $x$ approaches infinity, indicating that the corresponding series $ extbackslash sum_{n=1}^{ extbackslash infty} 1/n^p$ converges. However, for $p \leq 1$, the term $1/x^p$ does not approach 0 as $x$ approaches infinity, and the series $ extbackslash sum_{n=1}^{ extbackslash infty} 1/n^p$ diverges.
• Describe how the Integral Test can be used to determine the convergence or divergence of a series involving the term $1/x^p$.
  • The Integral Test can be used to establish the convergence or divergence of a series involving the term $1/x^p$ by comparing the series to the corresponding improper integral $ extbackslash int_1^{ extbackslash infty} 1/x^p \textbackslash, dx$. If the improper integral converges, then the series $ extbackslash sum_{n=1}^{ extbackslash infty} 1/n^p$ also converges. Conversely, if the improper integral diverges, then the series $ extbackslash sum_{n=1}^{ extbackslash infty} 1/n^p$ also diverges. The behavior of the improper integral $ extbackslash int_1^{ extbackslash infty} 1/x^p \textbackslash, dx$ is directly related to the value of $p$: if $p > 1$, the integral converges, and if $p \leq 1$, the integral diverges.
• Analyze the relationship between the convergence or divergence of the series $ extbackslash sum_{n=1}^{ extbackslash infty} 1/n^p$ and the corresponding improper integral $ extbackslash int_1^{ extbackslash infty} 1/x^p \textbackslash, dx$.
  • The convergence or divergence of the series $ extbackslash sum_{n=1}^{ extbackslash infty} 1/n^p$ is directly related to the behavior of the corresponding improper integral $ extbackslash int_1^{ extbackslash infty} 1/x^p \textbackslash, dx$. If the improper integral converges, then the series $ extbackslash sum_{n=1}^{ extbackslash infty} 1/n^p$ also converges. Conversely, if the improper integral diverges, then the series $ extbackslash sum_{n=1}^{ extbackslash infty} 1/n^p$ also diverges. This relationship is a consequence of the Integral Test, which allows us to determine the convergence or divergence of a series by comparing it to the corresponding improper integral. The value of $p$ is crucial in this analysis, as it determines the behavior of both the series and the improper integral.

Related terms

Power Function: A power function is a mathematical function in the form $f(x) = x^p$, where $p$ is a real number. The term $1/x^p$ is a specific type of power function where the exponent $p$ is positive.

Divergence Test: The Divergence Test is a method used to determine whether an infinite series converges or diverges. It is particularly useful for series involving the term $1/x^p$.

Integral Test: The Integral Test is another technique used to establish the convergence or divergence of an infinite series. The term $1/x^p$ plays a crucial role in the application of the Integral Test.