Reputation
27,504
Next tag badge:
76/100 score
16/20 answers
Badges
2 41 79
Newest
 Nice Answer
Impact
~437k people reached

Apr
18
comment Does every integer occur finitely many times and in what positions in Pascal's triangle?
Some links : Pascal triangle, "Singmaster's_conjecture", Singmaster's paper.
Apr
18
answered Find this integral $I(x)=\int_{0}^{+\infty}\frac{1}{y}e^{-y-\frac{x}{y}}dy$
Apr
17
comment Working out $\tan x$ using sin and cos expansion
Another thread of interest.
Apr
17
answered How to evaluate $\log x$ to high precision “by hand”
Apr
16
comment How to evaluate $\log x$ to high precision “by hand”
Yes you are right (about the accuracy and... that I'm assuming the answer! :-)). This told you could use one of the generalized c.f. for $\ln(2)$ given here (except that I didn't get quickly the correct fraction...). Another c.f. is here. Cheers,
Apr
16
comment How to evaluate $\log x$ to high precision “by hand”
A practical method to get the fraction $\dfrac {253}{365}$ is to start with the (supposed known) value $\ln(2)\approx 0.69315$ and to evaluate the continued fraction up to $ [0, 1, 2, 3, 1, 6, 3, 1]$ that is : $$\ln(2)\approx 0.69315\approx\cfrac 1{1+\cfrac 1{2+\cfrac 1{3+\cfrac 1{1+\cfrac 1{6+\cfrac 1{3+\cfrac 11}}}}}}=\frac {253}{365}$$
Apr
15
comment Question on finding formula for a Sequences
In case of confusion changing the point of view may help. If your point of view is global try for example to consider my derivations at a very low level (say by rederiving everything by hand...). If your confusion comes from the abstract $n$ replace it by a fixed value and so on. If confusions remain after all this ask very specific questions.
Apr
15
answered Question on finding formula for a Sequences
Apr
14
comment Question on finding formula for a Sequences
In fact this is one of the easiest example so consider only my second hint and think at what you obtained.
Apr
14
comment Question on finding formula for a Sequences
Another hint use this link to rewrite $\;y_t = 2^t + 2^{t-1} + .... 2^2 + 2 + 1\,$ as $2^{t+1}-1$.
Apr
14
comment Question on finding formula for a Sequences
$y_{n+1}=g(y_n,y_{n-1},\cdots)\,$ is a recurrence ; $t_n=f(n)$ is a 'general formula' or expression of whatever. The recurrence for $x_n$ should be simple enough to deduce the general formula.
Apr
14
comment Question on finding formula for a Sequences
Hint: since $y_{n+1}=2\,y_n+1\,$ search the corresponding recurrence for $x_n:=y_n+1$. Deduce the general formula for $x_n$.
Apr
12
comment Evaluation of the series $ \sum_{k=0}^{\infty}\frac{k}{2^{k+1}\left ( k+1 \right )^2}$.
To prove the second one use $\mathrm{Li}_2(z)=-\mathrm{Li}_2(1-z)-\log (1-z) \log (z)+\frac{\pi ^2}{6}$ with $z=\frac 12$.
Apr
12
comment Evaluation of the series $ \sum_{k=0}^{\infty}\frac{k}{2^{k+1}\left ( k+1 \right )^2}$.
Hint: rewrite the numerator as $(k+1)-1$ to get two series : the first one is a simple logarithm, the second will be the dilogarithm (both evaluated at $\dfrac 12$).
Apr
11
revised Is there a way to simplify a sum of cosecants?
Asymptotic expression for alternate sum.
Apr
11
awarded  Nice Answer
Apr
10
comment Is there a way to simplify a sum of cosecants?
@Claude and Apoapsis : In fact we can do much better as you may see in my updated answer... Cheers,
Apr
10
revised Is there a way to simplify a sum of cosecants?
Exact expansion. More terms...
Apr
3
comment Is there a way to simplify a sum of cosecants?
Thanks dear @Claude and no need to be sorry : you tried CAS expansions while I preferred wild numerical guesses. From the simplicity of $(7)$ we may hope some deeper (and proved!) results. Cheers,
Apr
3
revised Is there a way to simplify a sum of cosecants?
More complete answer.