Does the series 1 + 1/3 + 1/5 + 1/7 + ... diverge?

[happyg1]

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

 

[Muzza]

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.

 

[TD]

If you've seem comparison tests, try to find the general term for the series and compare it to the (divergent) harmonic series.

 

[happyg1]

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

 

[happyg1]

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