since -1 <= cosf(x) <= 1
and the second series diverge by integral test
Hello folks. First off i want to say this is an awesome forum! But lets get to the topic:
I totally understand the concept of convergence/divergence and i dont have problems to follow the lessons. But....
when it comes down to exercises/exams i totally fail in showing wether a series/sequence converges or not
i got the series:
my thinking was... finding a series a_n greater than the given one and then proof its convergence. but i seem to fail ....
another typical problem would be:
testing for convergence as n goes to infinity
my thought was writing the 1 as and then have raised to the power of 4n
but to be honest... im lost !
It is not possible to conclude.
Moreover, this kind of series comparison only works for series with a positive general term.
And since can be positive or negative, this method doesn't work here.
So we may use some things dealing with alternating series.
, positive sequences.
If , then
- if converges, converges
- if diverges, diverges
The tool for this one is the Abel transformation, which allows to prove the following theorem:
If decreases to 0, and is a sequence of complex numbers such that the partial sums are bounded (i.e. ), then the series converges.
In your case, and . In order to prove that satisfies the hypothesis, use the fact that and note that is a sum of terms in a geometric sequence.