calculus and analysis – Fourier Series of ODE

I am having trouble finding the Fourier series of a 2nd order ODE. Should I be using the piecewise function as well to set up the range for t?

Solve 𝑦′′ + 𝜔^2𝑦 = 𝑟(𝑡), where 𝑟(𝑡) = |𝑡|, -𝜋 < 𝑡 < 𝜋 using Fourier series

So far I have set up the ode and set equal to r(t)
r(t)=y''(t)+omega^2*y(t)
Plot(r(t),{t,-Pi,Pi})
Any help with the mathematica code would be greatly appreciated. How can I find An, Bn with the function being an ODE

Plotting with % works for fourier series but not with the function that produced %

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sequences and series – Lower bound for sum of reciprocals of positive real numbers

I am reading an article where the author seems to use a known relationship between the sum of a finite sequence of real positive numbers $a_1 +a_2 +… +a_n = m$ and the sum of their reciprocals. In particular, I suspect that
begin{equation}
sum_{i=1}^n frac{1}{a_i} geq frac{n^2}{m}
end{equation}

with equality when $a_i = frac{m}{n} forall i$. Are there any references or known theorems where this inequality is proven?

This interesting answer provides a different lower bound. However, I am doing some experimental evaluations where the bound is working perfectly (varying $n$ and using $10^7$ uniformly distributed random numbers).

Write a java program to find the sum of the following series: S=1+1/4!-2/9!+3/16!-4/25!…n terms [closed]

Sum of the series program
Conditions-PLs solve this program with scanner class
I have been trying to solve this program from many days but I am no able to, please help

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sequences and series – Determine which positive integers n have the following property:

This was asked by my professor.

Determine which positive integers $n$ have the following property: If $a[1] , dots, a[n]$ are $n$ real numbers greater than or equal to 1, and $A$, $G$, and $H$ are their arithmetic mean, geometric mean, and harmonic mean, respectively, then: $$G-H ≥ frac 1G – frac 1A$$

Time series classificatio

Hi

I have a question about classification of time series. Data has two features and I want to classify it into 5 classes. We have a stream of data and new data is generated every 5 seconds. Moreover in some classes we have inadequate data for training so we have classification problem with imbalanced data. I want to classify new data using machine learning methods according to the pattern shown in the figures. What methods do you suggest?

enter image description here

Sum of a series with

I am trying to compute the sum of the following series:
$$sum_{n=1}^infty 3^{n-1}sin ^3{frac{a}{3^n}}$$
The series is clearly convergent by the ratio test, as well as by comparison with $(frac{a}{3^n})^3$. If I take the derivative with respect to $a$, I am eventually left with $cos(frac{a}{3^n})-cos^3(frac{a}{3^n})$, which doesn’t seem very helpful, plus I’m not sure it’s okay to take the derivative. Multiplying by sine functions to try to get a telescopical sum didn’t lead me anywhere. Any ideas will be appreciated. Thanks!

How to show this series converges

I want to see if this series converges or not:
$$
sum_{n=1}^infty n^{-1/2}sin(n)sin(n^2).
$$

I tried comparision tests but nothing. I saw that integral criteria works but I don’t know how to show that.

Thank you

Prove the divergence or convergence of the following series.

I am trying to find whether the following series converges or diverges:

enter image description here