Series and Sequences ex
I'm preparing for my mathematics exams and am a little bit struggling with series and functions. Since my closest lecturer is 3000km away, decided to ask some help online:)
The following Series are convergent or divergent?
I solved it but really not sure if it is right.
And was struggling to solve this one for Sequences: whether or not this sequence converge, if does state limit: 2010^n / n^2010.
The second one is correct. Now for the first one
if , then it does not converge since the term oscillates more an more wildly.
If , then you can just use the alternating series test. This term goes to 0 as n goes to infinity, so we have that the series converges! The work you have shown is incorrect though, because you cannot factor -1 out of the absolute value. The limit is indeed 1, so it gives no information on convergence.
Thank You for the explanation and sorry for confusing you, it was (-1)^n in the example - bad paint skills!
Maybe have any ideas on the bottom exercise, the sequence 2010^n / n^2010 diverge ir converge ( limit ? )
The numerator is an exponential while the denominator is a polynomial. At first this ratio is very small for small numbers. However, when n = 2010, the ratio is 1. After that the numerator wins.
If you compare consecutive terms you have:
The limit of this is of course 2010 and since it's much larger than 1, the sequence diverges to infinity.