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Apr
23
comment 100-th derivative of a function
@Joel, Same as $$n=-1$$ right?
Apr
23
answered What is the geometry behind $\frac{\tan 10^\circ}{\tan 20^\circ}=\frac{\tan 30^\circ}{\tan 50^\circ}$?
Apr
23
answered n tends to infinity
Apr
23
comment Prove that $6(\sin^{10}A+\cos^{10}A) – 15(\sin^8A+\cos^8A) + 10(\sin^6A+\cos^6A) – 1 = 0​$
@HPDas, If $T_n=\cos^{2n}x+\sin^{2n}x$ $$T_0=2,T_1=1,T_{n+1}=T_n-\cos x\sin x\cdot T_{n-1}$$ To eliminate $\cos^2x,s=\sin^2x,$ $$6T_5-15T_4=-10T_3+1$$
Apr
22
comment Integral with irrational functions and polynomials
Use partial fraction decomposition, $$\dfrac y{1-y^3}=\dfrac A{1-y}+\dfrac{By+C}{1+y+y^2}$$ $$\implies y=A(1+y+y^2)+(1-y)(By+C)$$ Or $y^3-1=(y-1)(y-\omega)(y-\omega^2)$ $y=1\implies3A=1$ Constant $\implies0=A+C\iff C=-A $ Coefficients of $x^2\implies 0=A-B\iff A=B$
Apr
22
comment Prove that $ \tan^{-1}\frac{1}{3} + \tan^{-1}\frac{1}{7} + … + \tan^{-1}\frac{1}{n^2+n+1} = \tan^{-1}\frac{n}{n+2}$
See math.stackexchange.com/questions/1305576/… and math.stackexchange.com/questions/1346929/… and math.stackexchange.com/questions/1552649/…
Apr
22
comment Wolfram answer is different for the integral $\sqrt{\frac{x}{2-x}}dx$
@DavidH, Sorry for the typo
Apr
22
revised Wolfram answer is different for the integral $\sqrt{\frac{x}{2-x}}dx$
edited body
Apr
22
answered Wolfram answer is different for the integral $\sqrt{\frac{x}{2-x}}dx$
Apr
21
comment Show whether this trigonometry series converges
$$\le\dfrac{3n}{3^n}$$
Apr
21
awarded  Good Answer
Apr
21
comment Catalan numbers formula derivation
See en.wikipedia.org/wiki/Catalan_number#Proof_of_the_formula
Apr
21
comment 100-th derivative of a function
@Stephan. The real part of $e^{i(n\pi/4+x)}$ is $\cos(n\pi/4+x)$ So, the real part of $2^{n/2}e^x\cdot e^{i(n\pi/4+x)}$ is $2^{n/2}e^x\cos(n\pi/4+x)$ which is the required answer
Apr
21
awarded  Nice Answer
Apr
21
comment a/b + b/a is an integer if and only if a = b
@Hilly, what are the possible values of $k$ ?
Apr
21
comment a/b + b/a is an integer if and only if a = b
Observe that $k\pm r$ have the same parity
Apr
21
answered a/b + b/a is an integer if and only if a = b
Apr
21
comment Optimization of area of rectangle within semicircle
See math.stackexchange.com/questions/872223/…
Apr
21
comment Is the following solution correct?
If $x\ne\pm3i,$ we need $$x^2+1=x^2+9$$ which does not have a finite root
Apr
21
answered How to sum binomial coefficients which are multiples of 3?