julian's latest activity
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Jjulian replied to the thread Undergrad How to find the sum of this series?.@Rick16 asked how can the sum ##1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\cdots##be determined? In the following proof, the...
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Jjulian replied to the thread Graduate Series with rotating phase.You can prove ##\ln (1-z)## converges everywhere on the unit circle except at ##z=1## by Dirichlet's test: Fix ##z## in the unit circle...
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Jjulian replied to the thread Undergrad How to find the sum of this series?.We can obtain the relevant sum directly by considering the factorisation of ##\cos x## instead: \begin{align*} \cos x...
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Jjulian reacted to Matterwave's post in the thread Undergrad Problem in understanding composition of covariant derivatives with
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Could you explain your notation? Are you using abstract index notation? That is the notation I know that looks like what you used. In... -
Jjulian replied to the thread Undergrad How to find the sum of this series?.This is Euler's classic proof of the Basel problem that @.Scott was referring to in post #2. It appears in the book introduced by...
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Jjulian reacted to Rick16's post in the thread Undergrad How to find the sum of this series? with Like.
Thanks a lot. This is a derivation that I can follow. For physics purposes it is of course much easier to look up the zeta function and...
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Jjulian replied to the thread Undergrad How to find the sum of this series?.The particular derivation I gave requires knowledge of complex analysis and is quite lengthy. I therefore thought that it illustrated...
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Jjulian replied to the thread Undergrad How to find the sum of this series?.Extracting relevant bits from other thread: We can write \begin{align*} \sum_{n=0}^\infty \dfrac{1}{(2n+1)^2} & = \sum_{n=1}^\infty...
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Jjulian replied to the thread Undergrad How to find the sum of this series?.In another thread I provide a derivation of the general closed formula for ##\zeta(2k)## in terms of Bernoulli numbers: \begin{align*}...
