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Oct
21
comment Evaluating $\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$
@Anastasiya-Romanova: To perform the substitution $t\mapsto t^{1/c}$ on the second term in the integrand, we cannot simply break up the integral as $$\int_0^1\frac{t^{a-1}}{1-t}\mathrm{d}t-\int_0^1\frac{c\,t^{b-1}}{1-t^c}\mathrm‌​{d}t$$ since neither of those integrals converge. Thus, we need to compute $$\lim_{d\to1^-}\left[\int_0^d\frac{t^{a-1}}{1-t}\mathrm{d}t-\int_0^d\frac{c\,t^‌​{b-1}}{1-t^c}\mathrm{d}t\right]$$
Oct
21
revised Using Fourier Series to compute sums
fix typo
Oct
21
revised Evaluating $\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$
add another way to compute the integral
Oct
21
awarded  Guru
Oct
21
awarded  Necromancer
Oct
21
comment Slice of pizza with no crust
@Venus: I used Mathematica to make the animation.
Oct
21
awarded  Revival
Oct
21
revised Using Fourier Series to compute sums
fix typo
Oct
21
comment Using Fourier Series to compute sums
@Semiclassical: 1) $\beta(2n)$ doesn't have any known closed form either. 2) In this answer, I analogously develop a recursion for $\zeta(2n)$. Is that close?
Oct
21
answered Using Fourier Series to compute sums
Oct
21
comment Using Fourier Series to compute sums
@Semiclassical: I derive a recursion for $\beta(2n+1)$ in this answer. $\beta(2n)$ is a harder nut to crack, like $\zeta(2n+1)$.
Oct
21
revised Evaluating $\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$
add explanation
Oct
20
comment Evaluating $\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$
I've tried several values of $a$, $b$, and $c$ and they all match in Mathematica numerically to 30 places: $(1/2,1,3)$, $(1/2,2,3)$, $(1/3,2,3)$, and $(1/4,2,5)$.
Oct
20
comment Evaluating $\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$
@Chris'ssis: according to my computation, it's off by $\log(c)$.
Oct
20
answered Evaluating $\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$
Oct
20
comment The indefinite integral $\int{\frac{\mathrm dx}{\sqrt{1+x^2}}}$
@TedShifrin: if you stop answering elementary questions, who will improve the tone of the answers? It seems like leaving the sheep to the wolves.
Oct
20
comment Prove that $\sum_{n=1}^\infty \frac{n^2(n-1)}{2^n} = 20$
Please don't make nonessential changes to the question. One might think that you are simply trying to bring the question to the front page without really improving the question or adding a bounty.
Oct
19
comment how do I find the continued fraction of root n ??
@DigitalBrain: thanks! I have given a bit more explanation of what is being done in each line, too.
Oct
19
revised how do I find the continued fraction of root n ??
explain a bit more clearly what is being done in each line
Oct
19
comment How do I find the area of this region?
I used Intaglio, a Mac OS only drawing application.