Question

Let $P(z)=1+2z+3z^2+\cdots+nz^{n-1}$. Use the Gauss-Lucas theorem to show that all roots of this polynomial are in the unit disk.

score 1k · Answer 1 · 2026-03-30 14:42:06

1k

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Let $P(z)=1+2z+3z^2+\cdots+nz^{n-1}$. Use the Gauss-Lucas theorem to show that all roots of this polynomial are in the unit disk.

Published on 30 Mar 2026 - 14:42

score 167 · Answer 2 · 2018-04-10 04:13:29

167

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Find $\lim\limits_{n\to +\infty}\big(\frac{1}{\sqrt{n^2 + 1}} + \frac{1}{\sqrt{n^2 + 2}} + \cdots + \frac{1}{\sqrt{n^2 + n}}\big)$?

Published on 10 Apr 2018 - 4:13

score 278 · Answer 3 · 2026-03-31 10:07:38

278

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Use the Sequential Continuity Theorem to show that the function $f(x)$ is discontinuous at $x = 0$

Published on 31 Mar 2026 - 10:07

score 50 · Answer 4 · 2026-04-09 20:32:18

50

Views

Does this define a smooth map?

Published on 09 Apr 2026 - 20:32

score 160 · Answer 5 · 2026-04-12 05:29:34

160

Views

Question about derivative and duality product in a Banach space

Published on 12 Apr 2026 - 5:29

score 1.1k · Answer 6 · 2026-03-25 18:14:17

1.1k

Views

Partial Derivative Max and minimum

Published on 25 Mar 2026 - 18:14

score 32 · Answer 7 · 2026-03-29 14:25:06

32

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Prove by direct computation that $f^{[2]}(z) =\frac{1}{\pi i}\oint_{\partial D_1(0)} \frac{f(w)dw}{(w-z)^3} \text{ for } z \in D_1(0).$

Published on 29 Mar 2026 - 14:25

score 95 · Answer 8 · 2026-03-26 04:14:44

95

Views

Asymptotic behavior of the sequence $u_n=\int_0^1\frac{e^{-t}}{t+\frac{1}{n}}dt$

Published on 26 Mar 2026 - 4:14

score 24 · Answer 9 · 2026-04-11 18:49:52

24

Views

Does there exist only one such function up to constant?

Published on 11 Apr 2026 - 18:49

score 211 · Answer 10 · 2026-03-30 23:58:36

211

Views

Image of the boundary is equal to the boundary of image.

Published on 30 Mar 2026 - 23:58

score 394 · Answer 11 · 2026-04-11 12:58:46

394

Views

Convex function with unique critical point is coercive

Published on 11 Apr 2026 - 12:58

score 29 · Answer 12 · 2018-04-10 22:10:08

29

Views

If $f:\mathbb R \rightarrow \mathbb R$ is differentiable at $v$, why is $ x\rightarrow f(x-3)$ differentiable at $v+3$?

Published on 10 Apr 2018 - 22:10

score 323 · Answer 13 · 2026-03-30 23:56:47

323

Views

Prove that the set $\{x_n:n\in\mathbb{N}\}\cup\{a\}$ is sequentially compact using sequences

Published on 30 Mar 2026 - 23:56

score 409 · Answer 14 · 2026-04-02 20:21:12

409

Views

If $A\subset\mathbb{R}$ and A is countable then A is a union of countably many nowhere dense sets.

Published on 02 Apr 2026 - 20:21

score 88 · Answer 15 · 2026-03-27 16:28:12

88

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Let $B_n:=\sum_{k=0}^n f(k/n)B_{n,k}$. Show that $\lim_{n\to\infty}B_n(f)=f$

Published on 27 Mar 2026 - 16:28

List Question

Let $P(z)=1+2z+3z^2+\cdots+nz^{n-1}$. Use the Gauss-Lucas theorem to show that all roots of this polynomial are in the unit disk.

Find $\lim\limits_{n\to +\infty}\big(\frac{1}{\sqrt{n^2 + 1}} + \frac{1}{\sqrt{n^2 + 2}} + \cdots + \frac{1}{\sqrt{n^2 + n}}\big)$?

Use the Sequential Continuity Theorem to show that the function $f(x)$ is discontinuous at $x = 0$

Does this define a smooth map?

Question about derivative and duality product in a Banach space

Partial Derivative Max and minimum

Prove by direct computation that $f^{[2]}(z) =\frac{1}{\pi i}\oint_{\partial D_1(0)} \frac{f(w)dw}{(w-z)^3} \text{ for } z \in D_1(0).$

Asymptotic behavior of the sequence $u_n=\int_0^1\frac{e^{-t}}{t+\frac{1}{n}}dt$

Does there exist only one such function up to constant?

Image of the boundary is equal to the boundary of image.

Convex function with unique critical point is coercive

If $f:\mathbb R \rightarrow \mathbb R$ is differentiable at $v$, why is $ x\rightarrow f(x-3)$ differentiable at $v+3$?

Prove that the set $\{x_n:n\in\mathbb{N}\}\cup\{a\}$ is sequentially compact using sequences

If $A\subset\mathbb{R}$ and A is countable then A is a union of countably many nowhere dense sets.

Let $B_n:=\sum_{k=0}^n f(k/n)B_{n,k}$. Show that $\lim_{n\to\infty}B_n(f)=f$

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