Bounds on Gaussian infinite sum

What are some good upper and lower bounds on the following sum? S=+∞∑n=−∞1σ√2πe−12(nσ)2 I am looking for something better than 1<S<2. Answer If fact, there is an explicit solution to the expression S=+∞∑n=−∞1σ√2πe−12(nσ)2=1σ√2πϑ3(0,e−12σ2) where appears the elliptic theta function. The function decreases asymptotically to 1 but it goes to infinity for small values of σ. … Read more

Find a>1 s.t. ax=xa^x = x has a unique solution

What a makes {x∣ax=x} a singleton? (1.4444)x−x≤0 has real solutions. (1.4447)x−x≤0 has no real solutions. I guess 1.4444<a<1.4447 I tried running simulations using goal seek in Excel, but I think I’m doing it wrong because I keep getting a lot of values below 1.4. How can I approach this problem? Answer Note that ax=x⟺a=x1/x. So, … Read more

Montecarlo estimate of a integrand from 0 to ∞\infty

I have a question about monte carlo estimation of integrals. Suppose I am told to estimate using monte carlo, the integral: f(y)=∫y041+x2dx I want to estimate f(∞). I know that with some calculation, the exact values are given by π and 2π. However, there is some confusion with respect to defining the bounds and area … Read more

Why does $(\sin x)^2=x^2$ and $\sin x=x$ in these contexts?

Contexts (it must also be noted that as $\delta t$ tends to zero, $\delta \theta$ also tends to zero): First context $$ \lim_{\delta t \to 0} \frac{-2v\sin(\delta\theta/2)^2}{\delta t} = \lim_{\delta t \to 0} \frac{-2v(\delta\theta/2)^2}{\delta t}\quad\text{[since $\sin(x) \to x$ as $x \to 0$]} $$ Second context $$ \lim_{\delta t \to 0} \frac{v\sin(\delta\theta) – 0}{\delta t} = … Read more

Finding an approximation of a function’s root

I have the polynomial function f(x)=x5+2×2+1. I am trying to find an approximation to its root in [−2,−1], with the precision of 0.1, and with a minimal number of steps. The answer I was given was −17/16. I find it incorrect, and I wish to ask for your assistance. I have calculated f(−2) and f(−1), … Read more

How can I approximate this equation?

The approximation I’m having trouble with is this V=aln(1+La1−La)−2L, (a≫L) The hint was to use √1+x2=1+12×2+… and log(1+x)=x−12×2+… I couldn’t find a way to use the first hint, so I tried to use the second hint by spreading the ln function into two different terms, and using the approximation on each. Then I approximated all the … Read more

Given the approximation cos(x)≈1\cos(x)\approx 1, how small must xx be to have 12⋅10−8\frac{1}{2}\cdot 10^{-8} accuracy?

I have recently begun studying Numerical Analysis and have been given the following problem: For small values of x, how good is the approximation cos(x)≈1? How small must x be to have 12⋅10−8 accuracy? I am unsure what kind of answer is expected for the first question. For the second question I assume that I … Read more

A question about approximations.

We use approximations because they give a neat and clean answer and they are quite useful in physical situations too. For example, we can write: 51000+0.15 as 51000 , because both fractions are different only by 0.000000074988, which is quite small if we are making some calculations for physical situations. I’m having a doubt about … Read more

Approximating (n+1)Hn−nn2\frac{(n+1)H_n -n}{n^2}

I have the following value (n+1)Hn−nn2 But it is complicated to write, so I want to write a simple approximation to use it. I guess I can just write (n+1)Hn−nn2≈Hn−1n≈Hnn≈lnnn Is this a good approximation, or is there a clearly better one? Thanks. Answer Because γ>0.5, the approximation lognn is superior to γ+lognn despite γ=lim … Read more