please help with convergence divergence
Please look over the following series to determine wheather it is convergent or divergent by specific test. If my answer is correct, please just say the number of the problem and yes, if it is wrong, I would live to have an explanation, but if it's too much work, evn to know that it is wrong will help.
Thank you very, very much.
1) series infinity, n=1 (n^2-6)/(n+4) by N-th term test
My answer: lim of the n-th term does not exist, test failed
2) series infinity, n=1, 5^n/3^n (geometric series)
My answer: r=5/3 greater than 1, geometric series diverges.
3) series infinity, n=1, 1/n^(3/4) (p-series test)
My answer: p=3/4 less than 1, series diverges.
4) series infinity, n=1 (n^2-2n)/3n^3+4n^2) by limit comparison
My answer: both series a(n) and b(n) diverge based on b(n)
5) series infinity, n=1, 2^n/(2n+4)! by ratio test
My answer: lim 1/2n^2 =0 less than 1, converges
6) series infinity, n=1, (n-2)/(3n+4) by alternating test
My answer: diverges
7) series infinity, n=1, 1/(3n^4+1) by comparison test
My answer: converges
Thank you for your comments
I would greatly appreciate more explanation.
1. lim =n/1, which means lim does not exist. According to N-th term test, if lim does not equal to 0, then the series diverges, if lim = 0 then the test failed, series could eitherconverge or diverge.
How can I conclude that the series diverges? Is it correct to say, because lim does not equal to 0, the series diverges, even when lim does not exist?
4. a(n) is the original series, b(n) is a fraction discarded of all terms except those having the highest degree.
7. I compare it to b(n), which is 1/3n^4