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Some deleted answers! Some of them are accepted and the other are correct. I just would like to keep them at hand.

(1)- (2)- (3)- (4)- (5).

Is there a closed form for $\sum_{j=1}^{n} j^2\log{j}$?

$$ -\zeta' \left( -2 \right) +\zeta_1' \left( -2,n+1 \right). $$

A more general case

$$ \sum_{k=1}^{n} j^{s}\ln(j)^{m}= \lim_{\alpha \to s } \zeta^{(m)}_{\alpha}( - \alpha ) + \zeta^{(m)}_{\alpha} \left( -\alpha,n+1 \right). $$


Sep
12
comment Understand a Maple output
@RobertIsrael: I see.
Sep
12
comment Understand a Maple output
You want to solve for $L$ not $l$ I believe.
Sep
12
revised Why does this function decrease at this speed?
deleted 2 characters in body
Sep
12
revised Why does this function decrease at this speed?
edited body
Sep
12
comment Why does this function decrease at this speed?
@user119615: See my answer.
Sep
12
answered Why does this function decrease at this speed?
Sep
12
comment Why does this function decrease at this speed?
Is this the tail of a power series of an analytic function?
Sep
12
revised Compute $\sum\limits_{m=0}^{\infty}\frac{2^{m+1}}{i\lambda}(1+e^{-i\lambda\frac{2}{2^{m+1}}}-2e^{-i\lambda\frac{1}{2^{m+1}}})$
improved formatting
Sep
12
comment Show that if $X$ is a Hilbert space, then so is $Y$
Good job! Carry on.
Sep
12
comment Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$
@FelixMarin: Thanks for the comment. I really appreciate it.
Sep
12
comment Method of dominant balance and perturbation
See this.
Sep
12
comment Show that if $X$ is a Hilbert space, then so is $Y$
@user3784030: In $(1)$ It is already defined that $\langle y_1,y_2\rangle_Y=\langle x_1,x_2\rangle_X$. Just check the parallelogram law.
Sep
12
comment Show that if $X$ is a Hilbert space, then so is $Y$
@user3784030: The norm in Hilbert space is defined by the inner product. By the way you can do the same for $||y_1+y_2||$.
Sep
12
comment absolute convergence of an infinite series
@Leucippus: Thanks for the edit. I really appreciate it.
Sep
12
answered Show that if $X$ is a Hilbert space, then so is $Y$
Sep
12
answered absolute convergence of an infinite series
Sep
12
answered solving Legendre equation using the Frobenius method around a singular point
Sep
12
comment Is $\|T^2\|=\|T S\|$
Check the properties of the operator's norms.
Sep
12
comment Numerical analysis problem
You are asked to prove that it has exactly one root $(0,2)$.
Sep
12
comment Finding set of vectors that spans the solution set
@Valrok: You have two free variables and you are free how to choose them. In your case you chose $z$ and $w$.