Convergence Tests the Comparison Test

• Pixleateit
In summary, the comparison test states that if the larger series converges, then the smaller series will also converge and vice versa for divergence. However, there is no set method for finding the second series, it just requires some thinking and comparison to other known series. The alternating series test is different because the alternating series alternates between positive and negative terms, making it difficult to compare to other series.
Pixleateit
hey guys I'm kind of stuck on this idea of the comparison test. i understand the process and how it is done but i don't seem to understand how to find the second series(typically called bn) to compare with the original. get what i mean?

thanks! =)

Well the comparison test says that if you have $0\leq a_k \leq b_k$, then if

$$\sum_{k=1}^{\infty}b_k<\infty \quad \Rightarrow \quad \sum_{k=1}^{\infty}a_k <\infty$$

and if

$$\sum_{k=1}^{\infty}a_k=\infty \quad \Rightarrow \quad \sum_{k=1}^{\infty}b_k=\infty$$

So you need to find either a smaller $a_k$ or the larger $b_k$ in either case, but as you may see there isn't really any prescribed method of finding it. You just need to look at the series and think of what you could compare it to that is smaller or larger depending on the case. You can see why it's useful to know the convergence/divergence properties of many common infinite series.

so you can technically compare it to anything?

Yes, if after large $k$ the series fits the necessary condition of being larger or smaller. Of course this cannot be used for some obvious series which won't work, like alternating, which have their own tests.

just curious, why is the alternating series test different?

Well think of the alternating series, something like

$$\sum_{k=1}^{\infty} (-1)^n a_k$$

with $a_k$ always positive. This will jump back and forth from negative to positive on the number line as it progresses so most of the time a comparison test with these won't work (you would need to find something smaller or larger but you can see why this would be difficult).

What is the Comparison Test?

The Comparison Test is a method used to determine if an infinite series converges or diverges by comparing it to a known series with known convergence properties.

How do you use the Comparison Test to determine convergence?

To use the Comparison Test, you must first find a known series that is either larger or smaller than the given series. If the known series converges, then the given series must also converge. If the known series diverges, then the given series must also diverge.

What is the limit comparison test?

The limit comparison test is a variation of the Comparison Test that involves taking the limit of the ratio between the given series and the known series. If the limit is a positive, finite number, then the series will have the same convergence properties as the known series.

What are the conditions for using the Comparison Test?

The Comparison Test can only be used if the terms of the given series and the known series are all positive. Additionally, the known series must have known convergence properties and the terms of the given series must be less than or equal to the terms of the known series.

Can the Comparison Test be used for all infinite series?

No, the Comparison Test can only be used for some infinite series. It is most commonly used for series with positive terms, but there are some exceptions. It is important to check the conditions for using the Comparison Test before applying it to a series.

• Calculus
Replies
3
Views
1K
• Calculus
Replies
6
Views
1K
• Calculus
Replies
15
Views
3K
• Calculus and Beyond Homework Help
Replies
4
Views
512
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus
Replies
2
Views
1K
• Calculus
Replies
11
Views
2K
• Calculus
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
367
• Calculus
Replies
3
Views
1K