Finding Convergence Radius & Interval: Solving a Complex Homework Problem

So the final answer isIn summary, After applying the Ratio Test, the limit is -1/2. To get the radius and interval, the absolute value of -1 should be taken into account and the missing 1/2 in the denominator of the limit should be included. This solves the issue of getting a negative radius and the final answer should be (x+3)^3/2.
  • #1
Grunting7
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0

Homework Statement


∞
n=3 ∑ ((-1)n (x+3)3n)/(2nlnn)

Find radius of convergence, interval of convergence, values for x which series is: absolutely convergent, conditionally converge or divergence.

Homework Equations

The Attempt at a Solution


I applied the Ratio Test and got

|(x+3)3| lim n--> ∞ (-1(lnn))/(ln(n+1))

Then I used l'Hospitals to get the limit and got -1/2.
So then it's -1/2 * |(x+3)3| = L. Then do the radius and interval stuff.

The problem is that it's -1/2 and the radius can't be negative. I've had one or two similar problems where I keep getting a negative radius. Not sure what I am missing.
 
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  • #2
The absolute value of -1 is 1. When you're pulling out the (x+3)^3, you have to keep the absolute value on the lnn/ln(n+1) or pull out the -1 with the (x+3)^3. Also I am pretty sure you're missing (1/2) somewhere in your limit.
 
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Likes Grunting7
  • #3
Wow, did not even know that. Solves the negative radius I was having with the other problems.
Thanks alot!

Yea, I forgot the 2 in the denominator of the limit.
 

Related to Finding Convergence Radius & Interval: Solving a Complex Homework Problem

1. What is the purpose of finding convergence radius and interval?

The purpose of finding convergence radius and interval is to determine the values of x for which a complex function converges. This information is important for understanding the behavior of the function and for making predictions about its values at different points.

2. How do you calculate the convergence radius and interval?

To calculate the convergence radius and interval, you can use the ratio test or the root test. These are methods for determining if a series converges or diverges based on the values of its terms. The convergence radius is the value of x for which the series converges, while the convergence interval is the range of x values for which the series converges.

3. What are some common challenges when solving a complex homework problem?

Some common challenges when solving a complex homework problem include understanding the problem and its requirements, identifying the correct method or formula to use, and performing the necessary calculations accurately. Complex problems may also involve multiple steps and require a strong understanding of mathematical concepts.

4. How can I check if my solution for finding convergence radius and interval is correct?

You can check if your solution for finding convergence radius and interval is correct by plugging in different values of x within the convergence interval and seeing if the series converges or diverges. You can also compare your solution to the solutions of other students or to the solution in the textbook or class notes.

5. Are there any tips or tricks for solving complex homework problems?

One tip for solving complex homework problems is to break down the problem into smaller, more manageable parts. This can help you better understand the problem and identify the correct method to use. It is also helpful to practice solving similar problems and to seek help from a teacher or tutor if you are struggling with a specific concept or step.

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