Series Test for Convergence Problem

[porroadventum]

1. Is 1/(√(2n-1) convergent?


2. I have tried the first comparison test: a_n= 1/(√(2n-1) and b_n=1/(n^1/2. 0<=a_n<=b_n. But b_n diverges so we get no information.

I have tried the second comparson test and let b_n=1/n. But a_n/b_n=∞ so once again I get no information.

I have tried the ratio test but the lim_n→∞a_n+1/a_n=1 so I get no information.

I have tried the root test but limsup_n→∞a_n=1 so I get no information..



3. I have run out of options! Can anyone offer a hint as to what I should try next? Perhaps a different b_n for the first comparison test?

 

[gottfried]

I'm busy doing my first course in real analysis at the moment so I'm no expert and could well be wrong but given than the sequence is decreasing you can and probably should use the integral test.

 

[jbunniii]

1. Is 1/(√(2n-1) convergent?

I assume you mean
$$\sum_{n = 1}^\infty \frac{1}{\sqrt{2n-1}}$$
Note that $\sqrt{2n-1} \leq 2n-1 < 2n$. Try using this fact to apply the comparison test.

 

[porroadventum]

Sorry, I do mean the summation of 1/(√(2n-1)! Thank you very much for your help.

 

[SammyS]

1. Is 1/(√(2n-1) convergent?[/b]

2. I have tried the first comparison test: a_n= 1/(√(2n-1) and b_n=1/(n^1/2. 0<=a_n<=b_n. But b_n diverges so we get no information.

I have tried the second comparison test and let b_n=1/n. But a_n/b_n=∞ so once again I get no information.

I have tried the ratio test but the lim_n→∞a_n+1/a_n=1 so I get no information.

I have tried the root test but limsup_n→∞a_n=1 so I get no information.[/b].

3. I have run out of options! Can anyone offer a hint as to what I should try next? Perhaps a different b_n for the first comparison test?[/b]

(You may get a warning from a Moderator regarding posting in bold typeface.)


If the series diverges, then you need to compare your series to a divergent series which is smaller, term by term.

 

[porroadventum]

I did not know I was not allowed to post using bold. I noticed the bold input letters were already there so just typed within them... I'm sorry I will not do it again.

 

[Mark44]

Bolding removed. To the OP - please don't remove the parts of the template (problem statement, relevant equations, etc.). Removing these parts and then typing between the B tags caused everything you wrote to appear in bold.