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Aug
10
revised To find the closed form of $ f^{-1}(x)$ if $3f(x)=e^{x}+e^{\alpha x}+e^{\alpha^2 x}$
added 8 characters in body
Aug
9
comment To find the closed form of $ f^{-1}(x)$ if $3f(x)=e^{x}+e^{\alpha x}+e^{\alpha^2 x}$
@RobertIsrael : We know we can find a closed form $y''=y$ and $y(x)=\cosh x=(e^x+e^{-x})/2$ and $y^{-1}(x)=arccosh x=\int \frac{dx}{\sqrt{x^2-1}}$ .I wanted to go one more step and I wrote $y'''=y$.
Aug
9
revised To find the closed form of $ f^{-1}(x)$ if $3f(x)=e^{x}+e^{\alpha x}+e^{\alpha^2 x}$
deleted 2 characters in body
Aug
9
asked To find the closed form of $ f^{-1}(x)$ if $3f(x)=e^{x}+e^{\alpha x}+e^{\alpha^2 x}$
Aug
6
answered Can you prove why consecutive diagonal intersection points show decreasing fractions inside a rectangle?
Jul
31
answered I want to prove $ \int_0^\infty \frac{e^{-x}}{x} dx = \infty $
Jul
28
revised A question about complex exponential
deleted 5 characters in body
Jul
28
revised A question about complex exponential
edited body
Jul
28
answered A question about complex exponential
Jul
26
answered Is there a Definite Integral Representation for $n^n$?
Jul
25
accepted Is there a known closed form number for $\prod\limits_{k=2}^{ \infty } \sqrt[k^2]{k}$
Jul
25
comment Is there a known closed form number for $\prod\limits_{k=2}^{ \infty } \sqrt[k^2]{k}$
@RaymondManzoni : It is very nice way. I can see now I proved that $\int_0^{\infty}\int_x^{\infty} e^{-\gamma}\sum\limits_{n = 1 }^ \infty \frac{1}{n} \frac{d^n}{d\gamma^n}(\frac{1}{e^{\gamma}-1})d\gamma dx=\zeta'(2)$. Thanks a lot.
Jul
25
accepted What is the limit value of $a$ that $\sum\limits_{k = 1 }^ \infty \frac{ \ln(k)}{k^a}$ converges
Jul
25
comment What is the limit value of $a$ that $\sum\limits_{k = 1 }^ \infty \frac{ \ln(k)}{k^a}$ converges
Thank you very much for edit.
Jul
25
revised Is there a known closed form number for $\prod\limits_{k=2}^{ \infty } \sqrt[k^2]{k}$
edited body
Jul
25
comment What is the limit value of $a$ that $\sum\limits_{k = 1 }^ \infty \frac{ \ln(k)}{k^a}$ converges
Could you please add it to your answer? It would be perfect to see in your answer. Thanks a lot for your help.
Jul
25
asked Is there a known closed form number for $\prod\limits_{k=2}^{ \infty } \sqrt[k^2]{k}$
Jul
25
comment What is the limit value of $a$ that $\sum\limits_{k = 1 }^ \infty \frac{ \ln(k)}{k^a}$ converges
need LHospital rule to proof. Right?
Jul
25
comment What is the limit value of $a$ that $\sum\limits_{k = 1 }^ \infty \frac{ \ln(k)}{k^a}$ converges
Thanks for answer. Could you please add the prove of your first sentence in your answer?
Jul
25
asked What is the limit value of $a$ that $\sum\limits_{k = 1 }^ \infty \frac{ \ln(k)}{k^a}$ converges