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Apr
25
comment Is $\sum_{k=-\infty}^{0}f(-k)=\sum_{k=0}^{\infty} f(k)$ always?
@Alamos divergent series depend on the order
Apr
25
comment Is $\sum_{k=-\infty}^{0}f(-k)=\sum_{k=0}^{\infty} f(k)$ always?
It this true for all summation techniques used for divergent series?
Apr
25
comment Is $\sum_{k=-\infty}^{0}f(-k)=\sum_{k=0}^{\infty} f(k)$ always?
I made a typo in the question, sorry.
Apr
4
comment Does it make sense to learn any other language except English, being a mathematician?
@Howard Langtone consider Gelfond, Calculus of finite differences (1959). inis.jinr.ru/sl/vol1/UH/_Ready/Mathematics/… It has been translated to English only in 1971 in India. It is the only book where I found criteria of possibility to represent an analytic function as Newton series.
Apr
1
comment Has anybody ever considered “full derivative”?
@columbus8myhw this is not closed form...
Apr
1
comment Has anybody ever considered “full derivative”?
@columbus8myhw it is not equal to e.
Apr
1
comment Has anybody ever considered “full derivative”?
@columbus8myhw I think e can be expressed in closed form by modifying the formula.
Mar
28
comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$
@Martin R as $\log |a|$
Mar
28
comment Proof that $\lim_{x\to 0^+}{\sin \frac1x}=\sin \left(\frac{1}{2}\right)-\frac 12 \text{Ci}\left(\frac{1}{2}\right)$
@Timbuc en.wikipedia.org/wiki/Non-standard_analysis
Mar
28
comment Proof that $\lim_{x\to 0^+}{\sin \frac1x}=\sin \left(\frac{1}{2}\right)-\frac 12 \text{Ci}\left(\frac{1}{2}\right)$
@abel Cosine integral mathworld.wolfram.com/CosineIntegral.html
Mar
28
comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$
@Arpan Banerjee we get infinity in both numerator and denomenator with this rule. If u know how to apply it properly, make an answer please.
Mar
28
comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$
@Arpan Banerjee logarithm is not differentiable at 0, L'Hopital's rule is not applicable
Mar
28
comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$
@Arpan Banerjee counter-example: $x=1$, $\varepsilon$=2
Mar
28
comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$
@math also if $x-\varepsilon$ <0 the seond factor also becomes infinite, and I am exactly interested in the case $|x|<\varepsilon$...
Mar
28
comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$
@math are u sure? What if a tends to zero from below? then the second factor in the numerator becomes infinite...
Mar
28
comment Given this operator what is inverse operator?
@Martin R regarding sums to non-integer limits, look here: en.wikipedia.org/wiki/Indefinite_sum anyway, I found what I was looking for.
Mar
28
comment Given this operator what is inverse operator?
@Martin R 4 well the limit is added for some functions that have t in the denomenator (like 1/x). The sum should be computerd in closed form then take limit. The limit is not necessary for other functions.
Mar
28
comment Given this operator what is inverse operator?
@Olivier Oloa actually, $$\Delta_{sym}[f(x)]=(\Delta_{full}[f(x)]+\Delta_{full}[f(x-\varepsilon)])/2$$, but what does it help?
Mar
26
comment Why hyperreal numbers are built so complicatedly?
Differentiability: $$(f(x+\varepsilon)-f(x-\varepsilon))/(2\varepsilon)$$ What needs more definition with it?
Mar
26
comment Why hyperreal numbers are built so complicatedly?
Well what questions still remain unanswered with the definition from the question?