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Jun
26
comment $p$ and $r$ are primes greater than $2$. $p+r$ vs $p+2r$, which could be a prime number?
You need to find a divisor for $p+r$. Look at your example. What is the smallest (non trivial) divisor of $3+5$ ?
Jun
24
comment How do I derive $n!$ from this series?
Are you sure about that statement (it seems false) ? Where is it from ? Maybe give a bit more context.
Jun
24
comment Simplify P(n), where n is a positive integer : $ P(x)=\sum \limits_{k=1}^\infty \arctan\left(\frac{x-1}{(k+x+1)\sqrt{k+1}+(k+2)\sqrt{k+x}}\right). $
When evaluating $P(1)$, you should replaces instances of $x$ (and not $k$) with $1$. So you should get $$P(1) = \sum_{k = 1}^{\infty} \arctan \left(\frac{1-1}{(k+2)\sqrt{k+1} + (k+2)\sqrt{k+1}}\right) = \sum_{k = 1}^{\infty} \arctan(0) = 0$$
Jun
23
comment Sum of a series converges iff the sum of a function of the series converges
@hHhh : Ah yes, you're right, I was assuming that !
Jun
23
comment Sum of a series converges iff the sum of a function of the series converges
This doesn't work for any function $f$, but it does work for $f = \sin$. You need to look at how the function behaves close to $0$ (remember a series $\sum a_n$ converges depending roughly on how fast $a_n$ converges to $0$).
Jun
23
comment Finding $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{1}{1+\sqrt{\tan x}}dx$
Just a naïve suggestion : you may try changing the variable to $u = \sqrt{\tan x}$.
Jun
22
comment Integrating $\frac{1}{1+e^x}$
Alternatively, you may use the fact that $\frac{1}{1+e^x} = \frac{e^{-x}}{1+e^{-x}}$ and find a primitive for the right hand side.
Jun
22
comment A ring R is an algebra iff the ambient space $X$ is in R
Yes, I think that's enough.
Jun
22
answered Limit of zeta function in $x = 1$
Jun
22
comment Is this a homomorphism? Matrix homomorphism $\phi:\Bbb R\to GL_2(\Bbb R)$
@Icuttrees : You're welcome :)
Jun
22
answered Is this a homomorphism? Matrix homomorphism $\phi:\Bbb R\to GL_2(\Bbb R)$
Jun
22
comment Geometric series in closed form.
You may use the fact that $(-1)^k x^{2k-2} = -\left(-x^2\right)^{k-1}$.
Jun
18
reviewed Reject How is a symmetric group the subgroup of the group of isometries of three-dimensional space?
Jun
18
reviewed Approve Isomorphism between $\Bbb{R}^2 \times \Bbb{R}^2$ and $\Bbb{R}^2 \otimes \Bbb{R}^2$
Jun
11
comment Group isomorphism and matrices
If you look at the map $g \mapsto \psi(\phi_g)$ from $G$ to $T$ it's not injective, but if it's kernel is exactly $Z(G) = H$, then you get an injective map from $Inn(G) = G/Z(G)$ to $T$.
Jun
11
answered Why is $2^{32}-1=(2+1)(2^2+1)(2^4+1)(2^8+1)(2^{16}+1)?$
Jun
9
comment If for a prime p $1+\frac{1}{2}+\frac{1}{3}+\ldots + \frac{1}{p-1}=\frac{a}{b}$ then show that p divides a. Moreover if $p>3$ then $p^2$ divides a.
However, I think the proof is fixable by being a little more careful : the first line of the computation is actually valid in $\mathbb{Q}$. Then you should factor out $p$ in the expression $p \sum_{n= 1}^{p'} \frac{1}{n(p-n)}$ before doing the rest of the proof modulo $p$, which will give you the divisibility by $p^2$.
Jun
9
comment If for a prime p $1+\frac{1}{2}+\frac{1}{3}+\ldots + \frac{1}{p-1}=\frac{a}{b}$ then show that p divides a. Moreover if $p>3$ then $p^2$ divides a.
How can you prove a congruence modulo $p^2$ by working in $\mathbb{Z}/p\mathbb{Z}$ ? Besides, since you change the variable in the sum by taking inverse, why not do it in the first step directly ?
Jun
9
comment Proof of Wolstenholme's theorem
How can we show a congruence modulo $p^2$ by working in $\mathbb{Z}/p\mathbb{Z}$ ? It seems to me your computation only shows that your sum is $0$ modulo $p$. You can get that property by $\sum_{k \in \mathbb{Z}/p\mathbb{Z}^*} k^{-1} = \sum_{k \in \mathbb{Z}/p\mathbb{Z}^*} k$ and notice that $\sum_{k \in \mathbb{Z}/p\mathbb{Z}^*} k = 0$ when $p > 2$ because it is translation invariant.
Jun
9
comment prove that for every integer $a>0$ there is a unique representation $a=r*s^2$
Have you tried finding such a representation on actual integers (like say $3$, $4$, $5$, $\ldots$ $12$, $\ldots$ $17$, $18$, $19$, $20$), to get a sense of what is going on ?