Calculus II: Convergence of Series with Positive Terms

In summary: The solution clearly refers to Exercise 31, which you didn't include (I think), so you might want to take a look there.
  • #1
domabo
32
0

Homework Statement


https://imgur.com/DUdOYjE
The problem (#58) and its solution are posted above.

Homework Equations


I understand that I can approach this two different ways. The first way being the way shown in the solution, and the second way, which my professor suggested, being a Direct Comparison Test.

Since I don't know how to write in Latex ( I apologize)... here's an image of relevant tests.
https://imgur.com/F2vgRiS
as well as information pertaining to the specifics of the problem: https://imgur.com/PmRdsEa

The Attempt at a Solution


I can see how the solution works besides the initial step. I do not know where the 1/192 comes from.
 
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  • #2
The solution clearly refers to Exercise 31, which you didn't include (I think), so you might want to take a look there.
 
  • #3
Math_QED said:
The solution clearly refers to Exercise 31, which you didn't include (I think), so you might want to take a look there.

I did include it under specifics pertaining to the problem
 
  • #4
domabo said:

Homework Statement


https://imgur.com/DUdOYjE
The problem (#58) and its solution are posted above.

Homework Equations


I understand that I can approach this two different ways. The first way being the way shown in the solution, and the second way, which my professor suggested, being a Direct Comparison Test.

Since I don't know how to write in Latex ( I apologize)... here's an image of relevant tests.
https://imgur.com/F2vgRiS
as well as information pertaining to the specifics of the problem: https://imgur.com/PmRdsEa

The Attempt at a Solution


I can see how the solution works besides the initial step. I do not know where the 1/192 comes from.

Presumably you already know that ##\ln n < n^q## for any ##q > 0## and large enough ##n##, so just choose ##q## to give ##12 q - 9/8 <-1##, hence ##q < 1/96.## Any such ##q## will do, but ##q =1/192## gives a "nice" value to ##12 q - 9/8.##
 

Related to Calculus II: Convergence of Series with Positive Terms

1. What is the definition of convergence in calculus?

In calculus, convergence refers to the behavior of a sequence or series as its terms approach a certain limit. A sequence is said to converge if its terms get closer and closer to a specific value as the sequence progresses towards infinity. Similarly, a series is said to converge if the sum of its terms approaches a finite value as the number of terms increases.

2. How do you determine if a series with positive terms converges?

To determine if a series with positive terms converges, you can use the comparison test, ratio test, or root test. The comparison test compares the given series to a known convergent or divergent series. The ratio and root tests use the limit of the ratio or root of the terms in the series to determine convergence. If the limit is less than 1, the series converges. If it is greater than 1, the series diverges.

3. What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series where the terms are positive and the series converges. On the other hand, conditional convergence refers to a series where the terms alternate between positive and negative, and the series only converges when the terms are arranged in a specific order. In other words, absolute convergence guarantees the convergence of a series, while conditional convergence may or may not converge depending on the arrangement of its terms.

4. How does the integral test work to determine convergence?

The integral test is a method used to determine the convergence of a series by comparing it to the convergence of an improper integral. If the integral from 1 to infinity of the series terms converges, then the series also converges. On the other hand, if the integral diverges, then the series also diverges.

5. Can a series with positive terms diverge to infinity?

Yes, a series with positive terms can diverge to infinity. This can happen if the terms in the series do not approach zero as the number of terms increases. In this case, the series will continue to grow without bound, resulting in divergence. It is important to note that just because a series has positive terms does not guarantee convergence.

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