User:Proof2015sun

Sum

$$\sum_{k=1}^{\infty}\frac{k^7}{e^{k\pi}-1}+ 11^2(2^n-1)\sum_{k=1}^{\infty}\frac{k^7}{e^{2k\pi}-1}- 2^{n+8}\sum_{k=1}^{\infty}\frac{k^7}{e^{4k\pi}-1}=\frac{8+9\cdot{2^{2n}}}{32}$$

$$\sum_{k=1}^{\infty}\frac{k^7}{e^{k\pi}-1}- 11^2(2^n+1)\sum_{k=1}^{\infty}\frac{k^7}{e^{2k\pi}-1}- 2^{n+8}\sum_{k=1}^{\infty}\frac{k^7}{e^{4k\pi}-1}=\frac{8-9\cdot{2^{2n}}}{32}$$

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$$\sum_{n=1}^{\infty}\frac{n^3}{e^{n\pi}-1}+ 11(2^n-1)\sum_{n=1}^{\infty}\frac{n^3}{e^{2n\pi}-1}- 2^{n+4}\sum_{n=1}^{\infty}\frac{n^3}{e^{4n\pi}-1}=\frac{2^n+2}{48}$$

$$\sum_{n=1}^{\infty}\frac{n^7}{e^{n\pi}-1}+ 11^2(1-2^n)\sum_{n=1}^{\infty}\frac{n^7}{e^{2n\pi}-1}- 2^8(2^{2n}-2)\sum_{n=1}^{\infty}\frac{n^7}{e^{4n\pi}-1}=\frac{26-9\cdot{2^{2n}}}{32}$$

$$\sum_{n=1}^{\infty}\frac{n^7}{e^{n\pi}-1}+ 11^2(2^n+1)^2\sum_{n=1}^{\infty}\frac{n^7}{e^{2n\pi}-1}- 2^8(2^{2n}+2^{n+1}+2)\sum_{n=1}^{\infty}\frac{n^7}{e^{4n\pi}-1}=\frac{26+18\cdot{2^n}+9\cdot{2^{2n}}}{32}$$

$$\sum_{n=1}^{\infty}\frac{n^7}{e^{n\pi}-1}+ 11^2(2^n-1)^2\sum_{n=1}^{\infty}\frac{n^7}{e^{2n\pi}-1}- 2^8(2^{2n}-2^{n+1}+2)\sum_{n=1}^{\infty}\frac{n^7}{e^{4n\pi}-1}=\frac{26-18\cdot{2^n}+9\cdot{2^{2n}}}{32}$$

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$$\sum_{n=0}^{\infty}\frac{2^{n+1}}\cdot{\frac{2n-1}{2n+1}}=4-\pi$$

$$\sum_{n=0}^{\infty}\frac{2^{n+1}}\cdot\left({\frac{2n-1}{2n+1}}\right)^2=\sqrt{8+2\pi}$$

(1):

lcm ; lowest common multiple

$$s(1)=1$$

$$s(n)=s(n-1)+lcm\left(n,s(n-1)\right)+{(-1)^n}n$$

$$\lim_{n \to \infty}\frac{s(n)}{(n+1)!}=\frac{2}{e}$$

(2):

$$s(1)=1$$

$$s(n)=s(n-1)+lcm\left(n,s(n-1)\right)-{(-1)^n}n$$

$$\lim_{n \to \infty}\frac{s(n)}{(n+1)!}=1-\frac{2}{e}$$

(3):

$$s(1)=1$$

$$s(n)=s(n-1)+lcm\left(n,s(n-1)\right)-(-1)^n$$

$$\lim_{n \to \infty}\frac{s(n)}{(n+1)!}=\frac{1}{e}$$

(4):

$$s(1)=1$$

$$s(n)=s(n-1)+lcm\left(n,s(n-1)\right)+(-1)^n$$

$$\lim_{n \to \infty}\frac{s(n)}{(n+1)!}=1-\frac{1}{e}$$

(5):

$$s(1)=1$$

$$s(n)=s(n-1)+lcm\left(n,s(n-1)\right)+1$$

$$\lim_{n \to \infty}\frac{s(n)}{(n+1)!}=e-2$$

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(6):

$$s(1)=1$$

$$s(n)=s(n-1)+lcm\left(n,s(n-1)\right)-1$$

$$\lim_{n \to \infty}\frac{s(n)}{(n+1)!}=3-e$$

(1):

$$s(1)=1$$

$$s(n)=s(n-1)+lcm(n,s(n-1))+n$$

$$s(n)=(n+1)!-1$$

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(2):

$$s(1)=a$$ ; $$a \ge 1$$

$$s(n)=s(n-1)+lcm\left(n,s(n-1)-a+1\right)+1$$

$$\lim_{n \to \infty}\frac{s(n)}{(n+1)!}=f(a)$$

$$f(1)=e-2$$

$$f(2)=e-\frac{11}{6}$$

$$f(3)=e-\frac{13}{6}$$,...

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(3):

General

Where a is not a multiples of 3

$$s(1)=6k+a$$ ; $$k\ge 0$$

$$s(n)=s(n-1)+lcm\left(n,s(n-1)-6k-a+1\right)+1$$

$$\lim_{n \to \infty}\frac{s(n)}{(n+1)!}=k+f(a)$$

and

Where a is a multiples of 3

$$\lim_{n \to \infty}\frac{s(n)}{(n+1)!}=0.5k+f(a)$$

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see Eric Rowland and Benoit Cloitre