Radius of convergence using Ratio Test

Khan Academy

4 Sept 201407:18

EducationalLearning

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TLDRThe video script delves into the convergence of an infinite series function defined by the sum of terms \( \frac{x^{n+1}}{4^n \cdot n^4} \) from \( n=1 \) to infinity. It explores the expansion of the series for specific values of \( x \) and applies the ratio test to determine the function's convergence. The key takeaway is the radius of convergence, which is found to be within the interval \( |x| < 4 \), indicating that the series converges to finite values for these \( x \) values.

Takeaways

📚 The video discusses an infinite series function with terms of the form \( X^{N+1} / (4^N \cdot N^4) \).
🔍 The function is expanded for the first few values of N to illustrate its behavior.
🤔 The focus is on determining the values of X for which the function converges to finite values, and the concept of radius of convergence is introduced.
🧐 The ratio test is suggested as a method to determine the convergence of the series.
📈 The ratio test involves taking the limit of the absolute value of the (N+1)th term over the Nth term as N approaches infinity.
📝 The limit simplifies to \( |X| / 4 \) after canceling out common factors and focusing on the highest degree terms.
📉 For the series to converge, the limit must be less than one, which translates to \( |X| < 4 \).
📐 The radius of convergence is thus defined as 4, indicating the range within which X must lie for the series to converge.
📉 The interval of convergence is given as \( -4 < X < 4 \), which are the X values for which the original function converges.
🔑 The takeaway is that understanding the behavior of infinite series can be crucial for determining the convergence of complex functions.
📚 The importance of the ratio test in analyzing the convergence of series is highlighted, providing a clear method to apply in such scenarios.

Q & A

What is the given function defined by an infinite series?
-The function is defined by the infinite series from N equals one to infinity, where each term is X to the power of N plus one, divided by four to the power of N times N to the fourth.
What is the first term of the series when N equals one?
-When N equals one, the first term of the series is X squared over four times one to the fourth, which simplifies to X squared divided by four.
What happens when X is set to two in the series?
-When X is set to two, the term becomes X to the third power over four squared, which is 2 cubed divided by 16, resulting in 8/16 or 0.5.
How does the series change when X is three?
-When X is three, the term is X to the fourth over four to the fourth power times four to the fourth, which simplifies to four to the eighth power.
What is the purpose of the ratio test in this context?
-The ratio test is used to determine the values of X for which the series converges to finite values, by examining the limit of the absolute value of the ratio of consecutive terms as N approaches infinity.
What is the formula for the ratio test applied to this series?
-The ratio test formula applied here is the limit as N approaches infinity of the absolute value of the (N plus one)th term over the Nth term, which simplifies to the absolute value of X over four.
What does the limit of the ratio test formula tell us about the convergence of the series?
-The limit of the ratio test formula tells us that the series converges if the absolute value of X over four is less than one.
What is the radius of convergence for this series?
-The radius of convergence for this series is the absolute value of X being less than four, which means |X| < 4.
How can the radius of convergence be expressed as an interval of convergence?
-The interval of convergence is expressed as -4 < X < 4, indicating the range of X values for which the series converges.
What is the significance of the highest degree terms in determining the limit for the ratio test?
-The highest degree terms are significant because they dominate the limit as N approaches infinity, allowing us to ignore lower degree terms which approach zero in the limit.
How does the script simplify the expression for the ratio test?
-The script simplifies the expression by dividing the numerator and the denominator by N to the fourth, focusing on the highest degree terms, and then taking the limit as N approaches infinity.

Outlines

00:00

📚 Introduction to Infinite Series Function

This paragraph introduces an infinite series function defined by a series that starts from N=1 to infinity, with each term being X to the power of N+1 divided by (4^N * N^4). The paragraph provides examples for when N equals 1, 2, and 3, calculating the respective terms for X values of 1, 2, and 3. The main point of discussion is to determine the values of X for which the function converges to finite values, and the concept of the radius of convergence is introduced. The ratio test is suggested as a method to find the convergence criteria.

05:02

🔍 Applying the Ratio Test for Convergence

The second paragraph delves into applying the ratio test to determine the convergence of the series. It explains the process of taking the limit of the absolute value of the (N+1)th term over the Nth term as N approaches infinity. The calculation simplifies to finding the limit of (X * N^4) over (4 * (N+1)^4), which further simplifies to X/4 considering the highest degree terms. The ratio test indicates that the series converges if the limit is less than one, leading to the conclusion that the absolute value of X must be less than four. This establishes the radius of convergence as well as the interval of convergence for the series, which is -4 < X < 4.

Mindmap

Keywords

💡Infinite Series

An infinite series is the sum of the terms of an infinite sequence. In the video, the function is defined by an infinite series, where each term contributes to the overall value of the function. The series continues indefinitely, and the focus is on understanding when this series converges to a finite value.

💡Convergence

Convergence refers to the condition under which an infinite series approaches a finite limit as the number of terms increases. In the context of the video, the goal is to determine for which values of X the series converges to a finite value, ensuring the function is well-defined.

💡Radius of Convergence

The radius of convergence is the distance within which a power series converges. In the video, it is determined that the series converges for values of X within a certain interval, specifically when the absolute value of X is less than four. This radius helps identify the range of X values for which the series is finite.

💡Ratio Test

The ratio test is a method used to determine the convergence or divergence of an infinite series. It involves taking the limit of the absolute value of the ratio of successive terms. In the video, the ratio test is applied to the given series to find the values of X that result in convergence.

💡Limit

A limit is the value that a function or sequence 'approaches' as the input or index approaches some value. In the video, the limit of the ratio of successive terms is taken as N approaches infinity, which is crucial in applying the ratio test to determine convergence.

💡Absolute Value

The absolute value of a number is its distance from zero on the number line, without regard to direction. In the video, the absolute value is used to simplify the expressions and determine the conditions under which the series converges, particularly when considering the ratio test.

💡Power Series

A power series is a series of the form Σa_n*(X-c)^n, where each term is a power of X. The function in the video is expressed as a power series, and the focus is on finding the interval of X for which this series converges. Power series are essential in understanding many functions in calculus.

💡N-th Term

The N-th term of a series is the term corresponding to a specific index N. In the video, the expressions for the N-th and (N+1)-th terms are derived and manipulated to apply the ratio test. These terms are crucial in understanding the behavior of the series.

💡Numerator and Denominator

The numerator is the top part of a fraction, while the denominator is the bottom part. In the video, various manipulations of the series terms involve simplifying these fractions to apply the ratio test and find the limit. Understanding these parts helps in breaking down complex expressions.

💡Interval of Convergence

The interval of convergence is the set of all values of X for which a power series converges. In the video, it is found that the series converges for X values between -4 and 4. This interval provides a clear range within which the function defined by the series is valid.

Highlights

Introduction of an infinite series function defined by the sum of terms of the form \( X^{N+1} / (4^N \cdot N^4) \) from \( N = 1 \) to infinity.

Expansion of the series for specific values of \( X \) to illustrate the function's behavior.

Application of the ratio test to determine the convergence of the series.

Calculation of the limit for the ratio test by dividing the (N+1)th term by the Nth term.

Simplification of the ratio test expression by focusing on the highest degree terms.

Identification of the limit as \( \lim_{N \to \infty} |X \cdot N^4 / (4 \cdot (N+1)^4)| \).

Further simplification to find the limit as \( |X / 4| \).

Application of the ratio test result to conclude that the series converges if \( |X / 4| < 1 \).

Derivation of the radius of convergence as \( |X| < 4 \).

Expression of the interval of convergence for the function as \( -4 < X < 4 \).

Explanation of the significance of the radius of convergence in determining the function's finite values.

Demonstration of the method to apply the ratio test to an infinite series function.

Clarification on how to interpret the results of the ratio test for series convergence.

Highlighting the importance of considering absolute values in the ratio test calculations.

Illustration of the process to simplify the ratio test expression step by step.

Explanation of how the highest degree terms dominate in the limit calculation.

Presentation of the final condition for the series convergence in terms of \( X \).

Summary of the method to determine the interval of convergence for an infinite series function.

Transcripts

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Radius of convergence using Ratio Test

Takeaways

Q & A

What is the given function defined by an infinite series?

What is the first term of the series when N equals one?

What happens when X is set to two in the series?

How does the series change when X is three?

What is the purpose of the ratio test in this context?

What is the formula for the ratio test applied to this series?

What does the limit of the ratio test formula tell us about the convergence of the series?

What is the radius of convergence for this series?

How can the radius of convergence be expressed as an interval of convergence?

What is the significance of the highest degree terms in determining the limit for the ratio test?

How does the script simplify the expression for the ratio test?