Hi everyone,
I’ve been dealing with some sequence problems where I need to check if the sequence is increasing or decreasing. Usually, I just look at u_{n+1} — u_n, but for some complex functions, it gets messy.
I’ve been trying to use the derivative f'(x) (treating n as x) to check for monotonicity. It makes things much easier, but I’m a bit worried about the domain. Since f'(x) is for x ∈ R and a sequence is only defined for n ∈ N*, is this approach always rigorous? Or are there cases where this might fail?
For instance, if I have u_n = 2^n — an, what’s the best way to find the range of a so that the sequence is strictly increasing? Should I stick to the derivative or is there a better "competitive programming" way to handle this?
Any advice would be great. Thanks!
If $$$f(x)$$$ is monotonic then so is $$$u_n$$$. But the inverse is not necessarily true
See $$$u_n=\frac{\cos(2\pi n)}{n}$$$.
tks you!!!!! and could you suggest any alternative methods for determining the monotonicity (increasing or decreasing nature) of a sequence when calculating the difference proves too difficult?
Yes, this method can be used if computing derivatives is easier than computing differences. You're just extending a function from integers to reals, if its derivative has the same sign everywhere in a real interval (not just at integer values), it'll be monotonous on that interval whether in reals or integers.
its not necessarily true since the derivative is the change at that point (n) for the analytic continuation, but u care about the change over an interval of length 1, so it might be decreasing at that point but u may see an overall increase, like for example: cos(2pin)exp(n) if the derivative does not change sign yea ig then u can, u can also try breaking the sequence to see if it is increasing, like for n^2/(n^2+1) u can see at 0 it is 0, it has a finite limit and since numerator and denominator grow at the same rate is most probably is monotonically inc, which it is.
Could you suggest any alternative methods for determining the monotonicity (increasing or decreasing nature) of a sequence when calculating the difference proves too difficult?
look up ratio test, if an+!/an > 1 its inc, also, u can use the fact R is an ordered field, so an^3,logan,e^an all are inc if an is and vice versa
In competitive programming, a better way to check monotonicity of something is with induction, i.e. you might find something like this:
But for more math oriented problems it works, other than the case where $$$f(x)$$$ is not monotonic and $$$u_x$$$ is as others stated