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 $$$u_n=\frac{\cos(2\pi n)}{n}$$$.
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.