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23h
comment The number $\sum\limits_{n=-\infty}^{\infty} \frac{1}{2^{n^2}}$ is transcendental
I think the usual short argument for the transcendence of Liouville numbers breaks down, and that the transcendence degree is finite.
23h
comment Limit of Fraction
The exponent is $\frac{x}{1+x}$. Your method will still work for this, since for $x\gt M$ we have $(1+x)^{x/(1+x)}\lt 2^x$.
23h
comment The number $\sum\limits_{n=-\infty}^{\infty} \frac{1}{2^{n^2}}$ is transcendental
I think it is not a Liouville number, the "$1$'s" are not sparse enough.
23h
comment Limit of Fraction
You are welcome, yes it is $0$, since $e^x$ grows much faster than $x$.
23h
answered Limit of Fraction
1d
comment How can I try myself to solve exponential equations easily?
Using same bases for the exponentiation as much as possible is often useful. Here use $4^x=2^{2x}=(2^x)^2$.
1d
comment Statistical Dependency Transitivity
In the first example given my answer, $X_1$ and $X_2$ are dependent, as are $X_2$ and $X_3$, but $X_1$ and $X_3$ are independent. Verifications of the dependence/independence were not given, but they are straightforward.
1d
comment Numerical analysis-secant method
Use $f(x)=e^x-2$. The example is not really persuasive, since we need to know how to evaluate the exponential function.
1d
answered Deriving Euler's theorem from Fermat's little theorem
1d
revised How many sequences $(i_1,\ldots,i_d)$ of fixed length `d` of positive integers satisfy $\alpha\le i_1+\cdots+i_d\le\beta$?
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1d
comment How many sequences $(i_1,\ldots,i_d)$ of fixed length `d` of positive integers satisfy $\alpha\le i_1+\cdots+i_d\le\beta$?
I had typo/error. at the end, was quoting Stars and Bars for $\ge 0$. For $\ge 0$ the correct formula would be $\binom{k+d}{d}$. For $\ge 1$, which is what your question is about, (positive integers), it is something else. Will edit.
1d
comment Solve the equation $(tan θ − 2)(9 sin^2 θ − 1) = 0$
The sine is $\pm \frac{1}{3}$. The domain of the trig functions has not been specified, so there are infinitely many answers, tough to do a comma-separated list.
1d
comment Halting problem is solvable
If we think in terms of adding axioms, yes. The formal system within which one might prove things about Turing machines is not specified in your question, so there is some detail to fill in. One could use as formal system some sort of formal arithmetic, and encode Turing machines via some indexing (Godel numbering).
1d
revised evaluate $\lim_{n\to\infty}\sum_{r=1}^{n-1}\frac{e^{r/n}}{n}$
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1d
comment evaluate $\lim_{n\to\infty}\sum_{r=1}^{n-1}\frac{e^{r/n}}{n}$
For the modified question, the calculation is fully correct.
1d
answered Halting problem is solvable
1d
comment Expected value using indicator variables
You are welcome.
1d
answered evaluate $\lim_{n\to\infty}\sum_{r=1}^{n-1}\frac{e^{r/n}}{n}$
1d
comment Relation between a group's cardinality and number of subgroups
For the uncountable case, note that any element of the group generates a (cyclic) subgroup which is finite or countably infinite.
1d
answered Expected value using indicator variables