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trap101
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Determine the monotonicity and boundedness of the sequence.
1) 4n/ (4n2 + 1)22) 2n/ 4n + 1Question: I'm having a problem in knowing whether the approach I'm using is providing the right solutions.
in 1) I used the an+1/an and tried to compare their ratios. I end up with: 4n+4/ (4n2 + 8n + 5)1/2 . Now I know this will be less than 1 when I use a few "test values" such as n = 1, 2, etc. But how am I certain that the direction of the sequence won't eventually change?
If I take the derivative of the original sequence I end up with (after simplifying): 16n2 - 2n + 4 = 0. In that equation I can't find any critical points so is it safe to say that the sequence is always increasing based on that logic?In 2) I did the an+1/an approach and got it was decreasing. But only was I was able to conclude that was from putting in "test values" at the end again. This is when I simplified the ratio to: 2(4n+1)/ 4n+1 +1. I tried to find the derivative and get critical points but I couldn't find anything. What should I do in this case?
Thanks
1) 4n/ (4n2 + 1)22) 2n/ 4n + 1Question: I'm having a problem in knowing whether the approach I'm using is providing the right solutions.
in 1) I used the an+1/an and tried to compare their ratios. I end up with: 4n+4/ (4n2 + 8n + 5)1/2 . Now I know this will be less than 1 when I use a few "test values" such as n = 1, 2, etc. But how am I certain that the direction of the sequence won't eventually change?
If I take the derivative of the original sequence I end up with (after simplifying): 16n2 - 2n + 4 = 0. In that equation I can't find any critical points so is it safe to say that the sequence is always increasing based on that logic?In 2) I did the an+1/an approach and got it was decreasing. But only was I was able to conclude that was from putting in "test values" at the end again. This is when I simplified the ratio to: 2(4n+1)/ 4n+1 +1. I tried to find the derivative and get critical points but I couldn't find anything. What should I do in this case?
Thanks