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Jan
26
awarded  Necromancer
Jan
22
awarded  Nice Question
Jan
14
answered Proving that $\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$
Jan
13
comment Proving that $\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$
Try Ramanujan's Master Theorem with the sum for $J_n(x)J_m(x)$.
Dec
28
answered Factorial and exponential dual identities
Dec
17
awarded  Popular Question
Dec
13
comment Validity of Taylor's Series
@Mathematics Using complex analysis there is quite an elegant way. It turns out that the series for $f(z)$ centred at $z=z_0$ has a radius of convergence $R$ determined by the "distance" (i.e. magnitude of the difference) between $z_0$ and the nearest point to it such that $f(z_1)$ is not analytic. In the case of the logarithm, the only non-analytic point is $z_1=0$ which is a branch point. Thus, $R=|1-0| = 1$.
Dec
13
answered Validity of Taylor's Series
Dec
8
awarded  Notable Question
Dec
8
awarded  Popular Question
Nov
25
awarded  Popular Question
Oct
29
awarded  Notable Question
Oct
29
awarded  Popular Question
Oct
14
comment The proof that $\sum_{n=1}^{\infty} \frac{1}{n^k} $converges for all $k>1$.
Try the integral test.
Oct
7
answered Continuity on a Set
Aug
19
awarded  Nice Answer
Aug
7
comment Has anyone heard of this maths formula and where can I find the proof to check my proof is correct? $\sum^n_{i = 1}i + \sum^{n-1}_{i=1}i = n^2$
...And why the downvote? It's perfectly rigourous.
Aug
7
asked Error in ratio approximation
Aug
6
revised Has anyone heard of this maths formula and where can I find the proof to check my proof is correct? $\sum^n_{i = 1}i + \sum^{n-1}_{i=1}i = n^2$
deleted 5 characters in body
Aug
6
answered Has anyone heard of this maths formula and where can I find the proof to check my proof is correct? $\sum^n_{i = 1}i + \sum^{n-1}_{i=1}i = n^2$