#Domination Rule and the Limit Comparison Test

29 messages · Page 1 of 1 (latest)

undone patrol
#

Have a problem where I need to solve if a series converges or diverges using the domination rule and the limit comparison test. With the initial problem being my a_n, I want to find a b_n using the domination rule so I can use the limit comparison test. The problem I am having is I'm confused what my b_n would be?

image.png
summer vigilBOT
#
  1. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
  2. Wait patiently for a helper to come along.
  3. Once someone helps you, say thank you and close the thread with:
    +close
  4. Feel free to nominate the person for helper of the week in #helper-nominations
  5. Do not ping the mods, unless someone is breaking the rules.
  6. If you're happy with the help you got here, and the server overall, you can contribute financially as well:
undone patrol
#

Originally how I solved the problem was by concluding that from the domination rule my b_n would also be the same equation as a_n and then I used the series ratio test to confirm b_n converged and then proceeded to the limit comparison test but I don't think I used the domination rule correctly and did not get the right b_n.

short sequoia
#

Here a ratio test sounds a lot faster

undone patrol
short sequoia
undone patrol
#

I also think its weird which is causing me to doubt if I am even using the domination rule correctly since my a_n and b_n are the same

short sequoia
#

I agree with you that the tests in question are really not appropriate

undone patrol
#

It feels like a very roundabout way to get a simple solution

short sequoia
#

What about subbing 2^n with n^7

#

while it should be blatantly clear why exponential functions are greater than polynomial functions, I guess it might also be required to prove

undone patrol
#

Using n^7 works better and make more sense

short sequoia
#

we just know that the series of 1/n² converges

undone patrol
#

Since we know 1/n^2 converges couldn't we just plug that in for our b_n and use the limit comparison test (limit of a_n / b_n) which equals 0, doesn't that prove a_n converges?

short sequoia
undone patrol
short sequoia
#

Well, that's true yeah

#

I'm sorry I didn't study this in english so I looked it up

#

and it just didn't say any conclusion for a limit of 0

#

but i mean, if you think it's fair game, go for it

#

it's just that here a plain comparison sufficed

undone patrol
#

Yea, no worries, it seems logical to me. I just didn't know if I was using the domination rule correctly to get a b_n for the limit comparison. I'll go with n^7. Thank you for the help, I appreciate it

short sequoia
#

np man

undone patrol
#

+close