Question

Prove by induction on $n\geq1$ that $f^{(n)}(x)=n!a_n(1+x)^{\frac{1}{2}-n}$ for all $x\in(-1,\infty)$

score 67 · Answer 1 · 2026-03-26 03:13:31

67

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Prove by induction on $n\geq1$ that $f^{(n)}(x)=n!a_n(1+x)^{\frac{1}{2}-n}$ for all $x\in(-1,\infty)$

Published on 26 Mar 2026 - 3:13

score 52 · Answer 2 · 2018-04-01 18:03:41

52

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Show that $S_N(x + y) > S_N(x) + S_N(y) - 1$

Published on 01 Apr 2018 - 18:03

score 76 · Answer 3 · 2018-04-01 18:34:43

76

Views

Showing the Taylor expansion converges to the function

Published on 01 Apr 2018 - 18:34

score 313 · Answer 4 · 2026-03-26 03:13:27

313

Views

Define $f(x)=\sqrt{1+x}$ for all $x\in(1,\infty).$ Prove that the Taylor series converges to $f$ for all $x\in(0,1)$.

Published on 26 Mar 2026 - 3:13

score 81 · Answer 5 · 2026-03-22 19:41:43

81

Views

If $f \in C^\infty (\mathbb{R})$ such that $f^{(n)}=0$ for some $n$, then $f$ is a polynomial?

Published on 22 Mar 2026 - 19:41

score 35 · Answer 6 · 2026-03-20 01:46:37

35

Views

Prove that $r_{0,n}$ and $r_{1,n}$ are convex on $(-\infty,0)$.

Published on 20 Mar 2026 - 1:46

score 245 · Answer 7 · 2026-03-25 18:50:37

245

Views

Principal part of Laurent expansion of $\frac{1}{(\cos(z)-1)(z-1)}$ in annulus $1<|z|<2\pi$

Published on 25 Mar 2026 - 18:50

score 2.3k · Answer 8 · 2026-03-25 06:00:29

2.3k

Views

Remainder term in Taylor series for the complex exponential map

Published on 25 Mar 2026 - 6:00

score 37 · Answer 9 · 2018-04-02 17:43:00

With an appropiate substitution of the Taylor polynomial around $0$ give an estimation of $sin 1$ with the accuracy of $10^{-4}$.

score 42 · Answer 10 · 2026-03-26 03:13:32

42

Views

Prove that $|R_n|\leq{n}B|a_n||x|$ where $B=\max\limits_{t\in[0,x]}\left|\frac{(x-t)^{n-1}}{1+t}\right|(1+t)^{-1/2}$

Published on 26 Mar 2026 - 3:13

score 67 · Answer 11 · 2026-03-25 08:08:58

67

Views

Finding lower bound of summation in inequality

Published on 25 Mar 2026 - 8:08

score 59 · Answer 12 · 2026-03-20 11:40:56

59

Views

Problem regarding to solve "6 nonlinear dynamic system of first order differential equation" analytically ( approximated )

Published on 20 Mar 2026 - 11:40

score 7.8k · Answer 13 · 2026-03-26 03:12:32

7.8k

Views

Third degree Taylor series of $f(x) = e^x \cos{x} $

Published on 26 Mar 2026 - 3:12

score 135 · Answer 14 · 2026-03-26 11:17:58

135

Views

Closed form of the power series of $\sum_{n=0}^{\infty} \sin(n)\,x^n$ (Mathematica)

Published on 26 Mar 2026 - 11:17

score 106 · Answer 15 · 2026-03-25 08:07:42

106

Views

Is there an approximation to $\frac{1}{-\log(1-e^{-\beta})}$

Published on 25 Mar 2026 - 8:07

List Question

Prove by induction on $n\geq1$ that $f^{(n)}(x)=n!a_n(1+x)^{\frac{1}{2}-n}$ for all $x\in(-1,\infty)$

Show that $S_N(x + y) > S_N(x) + S_N(y) - 1$

Showing the Taylor expansion converges to the function

Define $f(x)=\sqrt{1+x}$ for all $x\in(1,\infty).$ Prove that the Taylor series converges to $f$ for all $x\in(0,1)$.

If $f \in C^\infty (\mathbb{R})$ such that $f^{(n)}=0$ for some $n$, then $f$ is a polynomial?

Prove that $r_{0,n}$ and $r_{1,n}$ are convex on $(-\infty,0)$.

Principal part of Laurent expansion of $\frac{1}{(\cos(z)-1)(z-1)}$ in annulus $1<|z|<2\pi$

Remainder term in Taylor series for the complex exponential map

With an appropiate substitution of the Taylor polynomial around $0$ give an estimation of $sin 1$ with the accuracy of $10^{-4}$.

Prove that $|R_n|\leq{n}B|a_n||x|$ where $B=\max\limits_{t\in[0,x]}\left|\frac{(x-t)^{n-1}}{1+t}\right|(1+t)^{-1/2}$

Finding lower bound of summation in inequality

Problem regarding to solve "6 nonlinear dynamic system of first order differential equation" analytically ( approximated )

Third degree Taylor series of $f(x) = e^x \cos{x} $

Closed form of the power series of $\sum_{n=0}^{\infty} \sin(n)\,x^n$ (Mathematica)

Is there an approximation to $\frac{1}{-\log(1-e^{-\beta})}$

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