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
for example:
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 !
But if something is less than a divergent series, it may as well be convergent or divergent.
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
This is not the simplest example by far! It is not exactly alternating, just "kind of".
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.