Intereting Posts

Expectation of an Absolute Value that is the Standard Normal?
Fourier transform of the indicator of the unit ball
Lebesgue measure on Riemann integrable function in $\mathbb{R}^2$
Determining possible minimal polynomials for a rank one linear operator
The proof of $\sqrt{2}$ is not rational number via fundamental theorem of arithmetic.
On the space $L^0$ and $\lim_{p \to 0} \|f\|_p$
How can we calculate the limit $\lim_{x \to +\infty} e^{-ax} \int_0^x e^{at}b(t)dt$?
Solution to 2nd order PDE
How to prove $|S||T|\leq |S \cap T ||\langle S, T\rangle|$?
Eigenvalues of Block matrices with known eigenvalues
Is there a clever solution to this elementary probability/combinatorics problem?
Solution of SDE $dX_t = \mu(t)X_tdt + \sigma X_t dW_t$
Metric is continuous, on the right track?
Frobenius Morphism on Elliptic Curves
Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$

Are there general rules that apply? For example:

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

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

- Why does this series $\sum_{n=0}^{\infty} \frac{(n!)^{2}}{(2n)!}$ converge?
- Cluster Point in Fréchet-Urysohn Space is Limit Point?
- Showing the following sequence of functions are uniformly convergent
- Convergence of tetration sequence.
- Prove the infinite sum $\sum_{k=0}^{\infty}{\frac{(2k-1)!!}{(2k)!!}x^{2k}}=\frac{1}{\sqrt{1-x^2}}$
- Slick proofs that if $\sum\limits_{k=1}^\infty \frac{a_k}{k}$ converges then $\lim\limits_{n\to\infty} \frac{1}{n}\sum\limits_{k=1}^n a_k=0$

etc.

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

- Does $\sum_{n=1}^\infty n^{-1-|\sin n|^a}$ converge for some $a\in(0,1)$?
- Is $2 + 2 + 2 + 2 + … = -\frac12$ or $-1$?
- Asymptotic (divergent) series
- Why does $1+2+3+\cdots = -\frac{1}{12}$?
- Counter example to theorem in complex domain
- Positive limit of sequence vs. positive terms
- If $X_n \stackrel{d}{\to} X$ and $c_n \to c$, then $c_n \cdot X_n \stackrel{d}{\to} c \cdot X$
- A sequence of random variables that does not converge in probability.
- Ramanujan Summation
- Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$?

**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 $y_n=n^2$.

- Problem with multivariable calculus: $\lim_{(x,y)\to (0,0)} \frac{x^3 + y^3}{x^2 + y}$
- Sum of reciprocals of squares – bounding
- Alternating group $\operatorname{Alt}(n)$, $n>4$ has no subgroup of index less than $n$.
- finite subgroups of PGL(3,C)
- Finitely generated projective modules are isomorphic to their double dual.
- Separable space and countable
- Proving that the algebraic multiplicity of an eigenvalue $\lambda$ is equal to $\dim(\text{null}(T-\lambda I)^{\dim V})$
- Number of distinct $f(x_1,x_2,x_3,\ldots,x_n)$ under permutation of the input
- Is $72!/36! -1$ divisible by 73?
- Graph of the function $x^y = y^x$, and $e$ (Euler's number).
- How to intuitively understand eigenvalue and eigenvector?
- Integer $2 \times 2$ matrices such that $A^n = I$
- What IS conditional convergence?
- Easiest way to solve system of linear equations involving singular matrix
- Fubini theorem for sequences