# Questions tagged [divergent-series]

Questions on whether certain series diverge, and how to deal with divergent series using summation methods such as Ramanujan summation and others.

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### Proofs involving manipulation of divergent series

Is this proof valid even though the harmonic series it is based on is divergent? Prove: $$\sum_{n=2}^\infty (\zeta(n)-1)= 1$$ Where $\zeta$ as in Riemann's Zeta function is summed over all natural ...
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### Numerically compute an oscillating series

I would like to compute in a numerically stable way an oscillating series. Imagine I have a signal $C(n)$, $n\in\mathbb{N}$ which decays exponentially with $n$ e.g. $C(n) = e^{-2n}$. Also, imagine I ...
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### A continuous function whose Fourier series diverges at $0$?

I read this at many places such as wiki or elsewhere quote : It is possible to give explicit examples of a continuous function whose Fourier series diverges at $0$. For instance, the even and $2π$-...
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### Proving convergence theorem on power series

When proving convergence of power series, one encounters the following term: $$\left( 1 + \delta \right)\frac{\sqrt[k]{|a_k|}}{\lim\sup_{n\to\infty}\sqrt[n]{|a_n|}}$$ where $\delta>0$, and the ...
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### Manipulating divergent series for practical applications

I have a series summation of the form $$\tag{1} S(x) = \sum_{i = -\infty}^{\infty} (-1)^{i}\left[\Phi\left(2ix\right) - \Phi\left((2i-2)x\right)\right],$$ where $\Phi(.)$ is the standard normal ...
### If $\{x_n\}$ is positive, decreasing and $\sum x_n=\infty,$ then $\sum x_ne^{-\frac{x_{n}}{x_{n+1}}}=\infty$
The problem is Suppose that $\{x_n\}$ is positive, monotonic decreasing and $\sum_{n=1}^\infty x_n=+\infty$. Prove that $$\sum_{n=1}^\infty x_ne^{-\frac{x_{n}}{x_{n+1}}}=+\infty.$$ This is a past ...
### Prove that a series is divergent, with $\epsilon,N$
I have this question: Prove that $a_n=\frac{(-1)^nn+1}{n+2}$ is divergent, without proving it by contradiction, or by using any theorems from the book. Prove it by using $\epsilon$ and $N$ notation. ...