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 Feb 3 comment Prove that if $\sum_{n=1}^ \infty na_n$ converges, then $\sum_{n=1}^ \infty a_n$ converges. @A.S. indeed unnecessary, but sufficient for this problem :) Feb 3 comment Prove that if $\sum_{n=1}^ \infty na_n$ converges, then $\sum_{n=1}^ \infty a_n$ converges. @Jill_Johnson by checking coefficients before each $S_k$, since $S_k$ only appears in $\frac{S_{k+1} - S_k}{k+1}$ and $\frac{S_{k} - S_{k-1}}{k}$ Feb 3 answered Prove that if $\sum_{n=1}^ \infty na_n$ converges, then $\sum_{n=1}^ \infty a_n$ converges. Feb 2 comment No normed space such that its dual is equivalent to $C^1[0,1],||,||_{\infty}$ Ok, so after you new edit, you need to prove that if $X$ and $Y$ are equivalent and $X$ is complete, then so is Y Feb 2 comment No normed space such that its dual is equivalent to $C^1[0,1],||,||_{\infty}$ Then you can prove for any normed space $X$, its dual is complete Feb 2 comment The matrix with entries $M_{ij} = \frac1{t_i+t_j}$ is positive semidefinite this may help Feb 2 answered sum of open balls in normed space Dec 3 comment Let $\small\mathbf H=\begin{pmatrix}a&b\\0&a\end{pmatrix}$ Show that $e^{\mathbf Ht}=e^{at}\small\begin{pmatrix}1&bt\\0&1\end{pmatrix}$ @bluemoon $\frac{na^{n-1}}{n!} = \frac{a^{n-1}}{(n-1)!}$. Sorry I have to catch a flight. I will be back tomorrow if it's still not clear Dec 3 answered How to show $\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$ Dec 3 answered Let $\small\mathbf H=\begin{pmatrix}a&b\\0&a\end{pmatrix}$ Show that $e^{\mathbf Ht}=e^{at}\small\begin{pmatrix}1&bt\\0&1\end{pmatrix}$ Dec 2 comment Find $\lim_{n\to \infty}\frac{n}{\sqrt[n]{n!}}$ @ClementC. $\log(x_n)$ conveges to 1, which means $x_n$ converges to $e$ Dec 2 revised Find $\lim_{n\to \infty}\frac{n}{\sqrt[n]{n!}}$ added 38 characters in body Dec 2 answered Find $\lim_{n\to \infty}\frac{n}{\sqrt[n]{n!}}$ Nov 27 reviewed Leave Open if $2f\left(\frac{x}{x^2+x+1}\right)=\frac{x^2}{x^4+x^2+1}$ then what is $f(x)$? Nov 27 reviewed Close Heine Borel Theorem $\iff$ Bolzano Weierstrass Theorem? Nov 26 awarded Necromancer Nov 12 comment Given a random variable $X$ when is possible to find a probability measure Q s.t $s.t \frac{dQ}{d\mathbb{P}}=X$? For each $A\in \mathcal{F}$, define $Q(A) = E^{\mathbb{P}}(X1_{A})$, then check all the conditions in the definition of probability measure Oct 13 awarded Yearling Oct 13 comment Creating an increasing function $g$ given a continuous function $f$, such that $|f(x)-f(y)| \leq |g(x)-g(y)|$ Intuitively, draw the curve of $f$ progressively from left to right, and each time the curve goes down, you inverse the direction such that it goes up by the same scale. The the final curve you get represents the function $g$ Oct 10 revised Two inequalities for $\left(\int_{a}^{b}|f(x)|^rdx\right)^{\frac{1}{r}}$ added 260 characters in body