Convergence or Divergence of a Series with Multiplication Terms?

In summary, the given series is a summation of products from k=1 to infinity. The format may be confusing, but it can be solved using the ratio test. The limit of the terms is not 2/3 or 1, and the use of "\sum" indicates a sum.
  • #1
fiziksfun
78
0
[tex]\sum[/tex][tex]\frac{1*3*5 ... (2k-1)}{1*4*7 ... (3k-2)}[/tex]

from k=1 to infinity

Does this series converge or diverge??

I have no idea where to begin, I don't understand it's format. Aren't series usually A+B+C ... but this is just multiplication ?!

? So ?? HELP!
 
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  • #2
Aren't series usually A+B+C ... but this is just multiplication ?!

It is a summation...maybe? You have indicated SOMETHING by using [tex]\Sigma[/tex]. However, your use of it is vague enough that I cannot tell whether you mean to sum say 1 + 1*3/1*4 + 3*5/4*7 + etc... or the entire thing is just one term.
 
  • #3
fiziksfun said:
[tex]\sum[/tex][tex]\frac{1*3*5 ... (2k-1)}{1*4*7 ... (3k-2)}[/tex]

from k=1 to infinity

Does this series converge or diverge??

I have no idea where to begin, I don't understand it's format. Aren't series usually A+B+C ... but this is just multiplication ?!

? So ?? HELP!
Are you saying you don't know what "[itex]\sum[/itex]" means? Obviously this IS a sum. Each of the "A", "B", and "C" being summed involves a product.

Do you remember a very simple theorem about when a sum does not converge?
 
  • #4
Does it diverge because the lim as k approaches infinity is 2/3 ?
 
  • #5
I recall having had something similar in Calculus II. Have you tried Ratio test? It's probably more approachable that way.
 
  • #6
fiziksfun said:
Does it diverge because the lim as k approaches infinity is 2/3 ?

No. The limit of the terms is NOT 2/3. Use the ratio test as JinM suggested.
 
  • #7
Is the limit 1?
 
  • #8
fiziksfun said:
Is the limit 1?

Don't make wild guesses. You aren't learning anything that way. All the question is asking is for convergence/divergence of the sum. Use the ratio test.
 

1. What is the difference between convergence and divergence?

Convergence refers to the coming together or meeting at a common point, while divergence refers to the spreading or moving apart from a common point.

2. How do you determine if a series or sequence is convergent or divergent?

To determine the convergence or divergence of a series or sequence, you can use various tests such as the ratio test, the root test, or the integral test. These tests evaluate the behavior of the series/sequence as it approaches infinity and can determine if it is convergent or divergent.

3. What is the significance of convergence and divergence in mathematics and science?

Convergence and divergence are important concepts in mathematics and science because they describe the behavior and properties of various mathematical functions, series, and sequences. They also have applications in fields such as physics, engineering, and economics.

4. Can a series or sequence be both convergent and divergent?

No, a series or sequence cannot be both convergent and divergent. It can only be one or the other. However, some series or sequences may be conditionally convergent, meaning they are convergent but only under certain conditions.

5. How do we use convergence and divergence in real-world applications?

Convergence and divergence have many real-world applications, such as in determining the limit of a function, analyzing the behavior of financial investments, and predicting the outcome of physical processes. These concepts also play a crucial role in the development of new theories and models in various fields.

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