Series for sqrt(2)*Pi

1)

sqrt(2)*Pi = 4 + 8*Sum_{n >= 0} (-1)^n/(16*n^2 + 32*n + 15). See A141759.

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2) 

In the following the Eisenstein summation convention is assumed: that is,

Sum_{n = -oo..oo} means Limit_{N -> oo} Sum_{n = -N..N}:

sqrt(2)*Pi = 4*Sum_{n = -oo..oo} (-1)^n/(4*n + 1).

More generally, it appears that for k >= 0, k not of the form 4*m + 1,

sqrt(2)*Pi = -sign(cos(Pi*(k - 3)/4)) * 4*(2^floor(k/2))*k! * Sum_{n = -oo..oo} (-1)^n/((4*n + 1)*(4*n + 3)*...*(4*n + 2*k + 1)) (verified up to k = 50).

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3)

The following series representations for sqrt(2)*Pi appear to be the first two entries in an infinite sequence of series representations:

sqrt(2)*Pi = (2^4)*Sum_{n >= 0} (-1)^n * (2*n + 1)/((4*n + 1)*(4*n + 3));

sqrt(2)*Pi = 512/105 - (2^6)*4!*Sum_{n >= 0} (-1)^n * (2*n + 3)/((4*n + 1)*(4*n + 3)*...*(4*n + 11)).

Conjecture 1: for k >= 0, sqrt(2)*Pi = A(k) + (-1)^k *2^(2*k+2)*(4*k)!*Sum_{n >= 0} (-1)^n * (2*n + 2*k + 1)/((4*n + 1)*(4*n + 3)*...*(4*n + 8*k + 3)),

where A(k) is a rational approximation to sqrt(2)*Pi.

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4)

The following series representations for sqrt(2)*Pi appear to be the first two entries in an infinite sequence of series representations:

sqrt(2)*Pi = 4 + (2^3)*Sum_{n >= 0} (-1)^n * (4*n + 1)/((4*n + 1)*(4*n + 3)*(4*n + 5)).

sqrt(2)*Pi = 1408/315 - (2^5)*5!*Sum_{n >= 0} (-1)^n * (4*n + 1)/((4*n + 1)*(4*n + 3)*...*(4*n + 13)).

Conjecture 2: for k >= 0, sqrt(2)*Pi = B(k) + (-1)^k * 2^(2*k+3)*(4*k + 1)!*Sum_{n >= 0} (-1)^n * (4*n + 1)/((4*n + 1)*(4*n + 3)*...*(4*n + 8*k + 5)),

where B(k) is a rational approximation to sqrt(2)*Pi.

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5)

The following series representations for sqrt(2)*Pi appear to be the first two entries in an infinite sequence of series representations:

sqrt(2)*Pi = 16/3 - (2^4)*3!*Sum_{n >= 0} (-1)^n/((4*n + 1)*(4*n + 3)*(4*n + 5)*(4*n + 7)).

sqrt(2)*Pi = 14848/3465 + (2^6)*7!*Sum_{n >= 0} (-1)^n/((4*n + 1)*(4*n + 3)*...*(4*n + 15)).

Conjecture 3: for k >= 0, sqrt(2)*Pi = C(k) - (-1)^k * 2^(2*k+4)*(4k + 3)!*Sum_{n >= 0} (-1)^n/((4*n + 1)*(4*n + 3)*...*(4*n + 8*k + 7)),

where C(k) is a rational approximation to sqrt(2)*Pi.

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6) 

Conjecture 4: for k >= 1, sqrt(2)*Pi = D(k) - (-1)^k * E(k)*Sum_{n >= 0} (-1)^n * P(n,k)/((4*n + 1)*(4*n + 3)*...*(4*n + 8*k + 1 )),

where P(n,k) = 8*(8*k + 1)*n^2 + (64*k^2 + 48*k + 2)*n, D(k) is a rational approximation to sqrt(2)*Pi and E(k) = 2^(2*k+4)*(4*k)!/(8*k*(32*k^2 + 1)).

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Peter Bala , Nov 05 2023