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awarded  Good Answer
Nov
30
revised Is there a sequence such that $a_n\to0$, $na_n\to\infty$ and $\left(na_{n+1}-\sum_{k=1}^n a_k\right)$ is convergent?
added 18 characters in body
Nov
30
answered Is there a sequence such that $a_n\to0$, $na_n\to\infty$ and $\left(na_{n+1}-\sum_{k=1}^n a_k\right)$ is convergent?
Oct
30
awarded  Yearling
Sep
21
answered Integrable slowly varying function
Aug
20
comment Prove the theorem on analytic geometry in the picture.
In higher dimensions, the analog is the Steiner formula which provides a polynomial expansion for the volume of the ``$t$-fattening'' of any convex set $K$ by the unit ball $B$: $$Vol_n(K+tB) = \sum_{j=0}^n \binom{n}{j} W_j(K)t^j ,$$ where the coefficients $W_j(K)$ are characterized by Hadwiger's valuation theorem.
Jul
31
comment Have I just discovered an easy way to square numbers?
@Anoneemus I think your insight is fun, thanks for sharing it. Seems to me it often can be faster and easier to remember to add $x$ and its successor than to remember to add $2x+1$.
May
26
awarded  Necromancer
Jan
8
awarded  Good Answer
Oct
30
awarded  Yearling
Jun
13
answered What was the book that opened your mind to the beauty of mathematics?
Jun
6
comment $p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x)$, then all the roots of $p_k(x)$ are real
@Edwin I believe $p_0(x)$ is just some arbitrary polynomial, and is taken here to be of degree $m$.
May
28
comment counterexample for ill posedness of the laplace equation
"Laplace's equation" would be most accurate.
May
24
comment Differentiable Path of Operators and their Inverses
I'd suggest to consult one of the classic texts on operator theory for the relevant theory: Dunford and Schwarz vol. 1, or Kato's Perturbation Theory for Linear Operators, say. I think I recall that the calculation that ${\mathcal P}$ is a projection involves reducing a double integral to a single one. Basically, the point is that the contour $\Gamma$ encircles all the spectrum of $A(t)$ except zero.
May
23
answered Differentiable Path of Operators and their Inverses
May
13
comment Proving Taylor's Theorem by integrating n times?
Perhaps this answer to a related question will help. In short: Apply the mean value argument to the n-fold integral first to pull out $f^{(n)}(\xi)$, then evaluate the n-fold integral with constant integrand.
Apr
23
awarded  Enthusiast
Apr
22
answered How prove this integral inequality $6\left(\int_{0}^{1}f(x)dx\right)^2\le 1+ 8\int_{0}^{1}f^3(x)dx$
Apr
17
comment Modification of Gronwall's Lemma
The proof of Lemma 1.3.3 in [201] looks messed up. Perhaps it's left over from some earlier draft of the book. It's not too difficult though to derive the result by an approach similar to the proof of Lemma 1.3.1: Let $\Phi$ be the right hand side and differentiate, etc.
Apr
16
revised Determining why $\int_{\partial R} z\ dz = 0$ and $\int_{\partial R} z\ dz = 0$ independently of Cauchy's Theorem for a Rectangle
fixed bug in endpoint evaluations