Intereting Posts

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Memorylessness of the Exponential Distribution
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Are there general rules that apply? For example:

(convergent series) + (divergent series) = (divergent series)

(convergent series) * (divergent series) = (convergent series)

- Dirichlet series
- How to check the real analyticity of a function?
- What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?
- How to find the limit of the sequence $x_n =\frac{1}{2}$, if $x_0=0$ and $x_1=1$?
- Prove that $\sum_{n=1}^\infty \ln\left(\frac{n(n+2)}{(n+1)^2}\right)$ converges and find its sum
- Convergence of $x_{n+1} = \frac12\left(x_n + \frac2{x_n}\right).$

etc.

Are there steadfast rules like this? Or does it vary depending on the specific series?

- To what extent is it possible to generalise a natural bijection between trees and $7$-tuples of trees, suggested by divergent series?
- How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?
- Suppose every subsequence of X converges to 0. Show that lim(X)=0
- Divergent Series Intuition
- Convergence of sets is same as pointwise convergence of their indicator functions
- How to show convergence of my generalized Fourier Series to the values I specify in the body of this question.
- Does $\displaystyle\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) dx\,$?
- How can I prove the convergence of a power-tower?
- Sequence of convex functions
- Determining precisely where $\sum_{n=1}^\infty\frac{z^n}{n}$ converges?

**For the addition :** let $\sum u_n$, $\sum v_n$ and $\sum (u_n+v_n)$.

- If two of these series converge the last converges.
- If one converges and another diverges then the last diverges.
- If two diverge, we can’t say anything about the last. (e.g. $\sum n$

and $\sum (-n)$).

You can easily adapt this for the substraction.

**For the Cauchy product :**

- If $\sum u_n$ and $\sum v_n$ are absolutely convergent then the

Cauchy product is absolutely convergent. - If one is absolutely convergent and the other convergent, the Cauchy

product is convergent (Mertens’ theorem). - If they are (conditionally) convergent you can’t say anything (e.g.

$u_n=v_n=\frac{(-1)^n}{\sqrt{n+1}}$ and

$u_n=v_n=\frac{(-1)^n}{n+1}$). - Nevertheless, if $\sum u_n$, $\sum v_n$ and their Cauchy product are

convergent, then the Cauchy product is equal to $(\sum u_n)(\sum

v_n)$. - We can’t say anything about the Cauchy product of divergent series

(Think of power series).

**A quick summary :**

If $E$ is a Banach space (to have absolute convergence $\Rightarrow$ convergence) : the set $\mathcal S_C$ of convergent series with terms in $E$ is a vector space and the set $\mathcal S_{AC}$ of absolutely convergent series is a vector subspace of $S_C$.

Moreover, if $E=\mathbb R$ or $E=\mathbb C$, $S_{AC}$ has a ring structure.

If you want to consider $\sum u_nv_n$ instead of the Cauchy product you can use the Dirichlet’s test or an Abel transformation.

A strange operation is to permut terms : if $\sum u_n$ is absolutely convergent then $\sum u_{\varphi(n)}$ too, and $\sum u_n=\sum u_{\varphi(n)}$. But if $\sum u_n$ is conditionally convergent the result is false. Worse, $\forall S\in\mathbb R\cup\{+\infty,-\infty\}$, you can find a permutation $\varphi$ such as $\sum u_{\varphi(n)}=S$.

Another useful operation is to group terms : let $(p_n)_{n\in\mathbb N}$ a (strictly) increasing sequence with $p_n\in\mathbb N$. Let $v_0=\displaystyle\sum_{i=0}^{p_0}u_i$ and $v_n=\displaystyle\sum_{i=p_{n-1}+1}^{p_n}u_i$.

If $\sum u_n$ converges, then $\sum v_n$ converges too and $\sum u_n=\sum v_n$.

But, we usually can’t say anything if $\sum u_n$ diverges : take $u_n=(-1)^n$ and define $v_n$ by grouping two following terms $v_n=1-1=0$, then $u_n$ diverges but $v_n$ converges.

(There are still some results, but the answer would be too long).

**Conclusion :** take care when manipulating series. If I remember well, Euler made this error : let $S=1-1+1-1+\cdots=1-(1-1+1-\cdots)=1-S\Rightarrow S=\frac{1}{2}$ (that would be true if we used the Cesaro limit of partial sums) or maybe $S=1+2+4+8+\cdots=1+2(1+2+4+\cdots)=1+2S\Rightarrow S=-1$ (it would be true for the field of 2-adic numbers, but Newton didn’t know that).

The first rule you mention is true, as if $\sum x_n$ converges and $\sum y_n$ diverges, then we have some $N\in\mathbb N$ such that $\left|\sum\limits_{n=N}^\infty x_n\right|<1$ so $\left|\sum\limits_{n=N}^\infty (x_n+y_n)\right|\geq \left|\sum\limits_{n=N}^\infty y_n\right|-1\to\infty$. However, the second is false, even if the series converges to $0$. An easy example is when $x_n=1/n^2$ and $y_n=n^2$.

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