Using O-notation for asymptotic estimation of the number of additions in recursive function

the number of additions that are executed during the calculation is a(n). How can i find an asymptotic estimation for the function mystery(n) with the help of the O-notation and master theorem. Note: here the question is not asked for the value of mystery(n), but rather for the number of additions!

def mystery(n):     if n==0:         return n * n     return 2 * mystery(n/3) + 4 * n 

What is the asymptotic bound for $1n + 2(n-1) + 3(n-2) + … + (n-1)2 + n$?

My best guess is that the series $ $ \sum_{i=1}^n i(n-(i-1)) $ $ becomes $ $ 2 \Bigg[ n + 2(n-1) + … + \frac{n}{2} \bigg(n-\bigg(\frac{n}{2}-1\bigg)\Bigg)\Bigg] $ $ So the highest term is $ n^2$ and there are $ n$ terms. Does that mean its $ O(n^3)$ ? That seems high. Intuitively, it seems like it should be closer to $ O(n^2)$ but I can’t find a way to bring it down mathematically.

Asymptotic analysis of a complex expression.

$ A = -\frac{((a + k_{3}(\alpha + t_{1}v)^n c^n(\eta + 1) – 2)^2 – (a-1)^2 + 4l_{6}(\eta + 1)b – 1)c^n}{4l_{6}c^{2n}}$

$ a = k_{3}(\eta + 1)b$

$ b = ((\alpha + t_{1}v)*c)^n$

$ c = u_{1}v + \frac{1}{\alpha}$

I am trying to simplify this expression $ A$ taking $ n$ to be large and $ \eta$ ,$ \mu$ ,$ v$ to be very small. The calculation is getting tedious. Any other way of seeing this or source where I can put this and obtaina simplified expression?

Recurrence relation for the asymptotic expansion of an ODE

I want to solve for the asymptotic solution of the following differential equation

$ $ \left(y^2+1\right) R”(y)+y\left(2-p \left(b_{0} \sqrt{y^2+1}\right)^{-p}\right) R'(y)-l (l+1) R(y)=0$ $

as $ y\rightarrow \infty$ , where $ p>0$ . I did the standard way by obtaining a series solution by the Frobenius method prescription in the form

$ $ R(y)=\sum_{n=0}^\infty \frac{a_{n}}{y^{n+k}}$ $ where $ k=l+1$ is the indicial exponent. I had difficulty finding for a recurrence relation for the coefficients $ a_n$ for arbitrary value of the parameter $ p$ . Right now, I am just doing the brute force method of solving individual $ a_n$ for every value of $ p$ . But I am just wondering whether the recurrence relation is possible to solve. Any help is appreciated.

Asymptotic rate for the expected value of the square root of sample average

I have iid random variables $ X_1, \dots, X_n$ with $ X_i \geq 0$ , $ E[X_i]=1$ and $ V[X_i] = \sigma^2$ . Let $ S_n = \frac{\sum_{i=1}^n X_i}{n}$ .

I’d like to say that $ E[\sqrt{S_n}] = 1-O(1/n)$ .

My first approach was to write $ E[\sqrt{S_n}] = \sqrt{E[S_n] – V[\sqrt{S_n}]} = \sqrt{1-V[\sqrt{S_n}]}$ .

I’m then left with showing that $ V[\sqrt{S_n}] = O(1/n)$ .

I’m unsure how to go about this. First, can I hope to prove such an asymptotic bound in general? If not, are there extra assumptions that can be made on the $ X_i$ so that this holds true?

Asymptotic complexity of function with two Input variables

Suppose I have a function with two input below.

$ f(m,n) = \log {n^m} + 100n \log \log {m^5} + 150m + 4n^2 + 1000$ .

Is it safe to say that $ f(m,n)$ is $ \mathcal{O}(m \log n)$ , or is it $ \mathcal{O}(n^2)$ instead? I think the first one is more representative as it includes the variable $ m$ . But that may not be the case if $ n$ is relatively very larger than $ m$ .

Adiabatic limit for metric asymptotic to the Taub-NUT metric

Let $ (M^4,g)$ be a Ricci-flat Riemannian manifold asymptotic to the Taub-NUT metric. That is, there exists a circle fibration $ \pi:M\backslash K \to \mathbb R^3\backslash B(0,R)$ which is the Hopf fibration if restricted on $ S^2$ . Moreover, the metric $ g$ satisfies $ $ g=h+O(r^{-\tau}) ,\quad \nabla ^{h,k} g=O(r^{-\tau-k}) $ $ for some $ \tau>0$ and any integer $ k \ge 1$ , where $ h$ is the Taub-NUT metric and $ r=\sqrt{x_1^2+x_2^2+x_3^3} \circ \pi$ .

Now we define $ D_{s}=\{x\in M \mid r(x) \le s\}$ and consider the Hirzebruch signature formula: $ $ \tau(M)= \frac{1}{12 \pi^2} \int_{D_{s}} |W_+|^2-|W_-|^2\, dV+I(s)+\eta(\partial D_s) $ $ where $ I(s)$ involves the boundary integral of the second fundamental form and $ I(s) \to 0$ if $ s \to \infty$ .

Can we prove that $ \lim_{s \to \infty}\eta(\partial D_s)=\frac{2}{3} $ ?