Proving Divergent Series: 1 + 1/3 + 1/5 + 1/7 +...

In summary, the conversation discusses ways to prove that the infinite series 1+ 1/3 + 1/5 + 1/7 + 1/9 + ... diverges. Suggestions are made to compare the series to a divergent series or to group the terms in a way that shows they are greater than a certain fraction. The conversation also mentions the general formula for the series and the use of comparison tests. However, the person involved is struggling to find a solution and is seeking additional hints.
  • #1
happyg1
308
0
Hi,
I need to show that the infinite series 1+ 1/3 + 1/5 + 1/7 + 1/9 + ... diverges.

Am I correct in saying that it is a subsequence of the divergent harmonic series, therefore diverges?
Is there some other more elaborate (and correct) way of grouping the terms to show that they are greater than some fraction? Like the harmonic series has groupings that are all 1/2, so you get 1/2 + 1/2 + 1/2 +...


Thanks
CC
 
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  • #2
My hint for you: show that the given series is greater than some other divergent series. Remember that, say, 1/2 times a divergent series will also be divergent...

Am I correct in saying that it is a subsequence of the divergent harmonic series, therefore diverges?

Afraid not. Exercise: find a non-convergent sequence which has a convergent subsequence.
 
  • #3
If you've seem comparison tests, try to find the general term for the series and compare it to the (divergent) harmonic series.
 
  • #4
Hi,
For the general formula, I got SUM 1/(1+2n). We haven't studied any of the tests yet, so I don't know if my professor would accept the comparison test. I'm trying (unsuccessfully) to somehow group the terms of the sequence of partial sums to get them bigger than some number. So far I have done this:

1 + 1/3 + 1/5 + 1/7 + ... is equivalent to:
1/M + 1/(M+2) + 1/(M+2(2)) + ...+1/(M+2(k)) where k>M and M starts at 1.
If you guys have any pointers, please let me know. I feel like I'm getting close, but I can't see the way.
CC
 
  • #5
Hi,
My professor says to try to get 1/M + 1/(M+2) + 1/(M+2(2)) + ...+1/(M+2(k)) when k>M to be less then 1/4. And forget about the M=1. I guess M would need to be odd.

I'm SO tired of this question. If anyone knows anything, Please give me a hint.
CC
 

1. What is a divergent series?

A divergent series is a mathematical series in which the terms do not approach a finite limit as the number of terms increases. This means that the sum of the series will continue to increase without bound.

2. Can the series 1 + 1/3 + 1/5 + 1/7 + ... be proven to be divergent?

Yes, this series can be proven to be divergent using various mathematical methods such as the comparison test, the ratio test, or the integral test.

3. How does the comparison test prove the divergence of this series?

The comparison test is a method used to determine the convergence or divergence of a series by comparing it to another series with known convergence or divergence. In this case, the series 1 + 1/3 + 1/5 + 1/7 + ... can be compared to the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... which is known to be divergent. Since the terms in our series are always greater than or equal to the terms in the harmonic series, our series must also be divergent.

4. Can the ratio test be used to prove the divergence of this series?

Yes, the ratio test can also be used to prove the divergence of this series. The ratio test compares the absolute value of the terms in the series to the absolute value of the corresponding terms in the geometric series. In this case, the geometric series with a common ratio of 1/2 is known to be divergent, thus the ratio test can be used to prove the divergence of our series.

5. Why is it important to prove the divergence of a series?

Proving the divergence of a series is important because it allows us to know that the sum of the series will increase without bound. This information is useful in various mathematical applications, such as determining the convergence of integrals or solving differential equations.

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