Verified answer

The Binomial Series is given by (1+t)^a = 1 + Σ(n=1 to ∞) C(a,n) x^n,

where C(a, n) = a(a-1)...(a-n+1)/n!.

Letting a = -1/2 and x = t^2, we obtain

(1 + t^2)^(-1/2) = 1 + Σ(n=1 to ∞) C(-1/2, n) (t^2)^n

==> √(1 + t^2) = 1 + Σ(n=1 to ∞) C(-1/2, n) t^(2n).

[Here, C(-1/2, n)

= (-1/2)(-3/2)...(-1/2 - n + 1)/n!

= (-1/2)(-3/2)...(-(2n-1)/2)/n!

= (-1)^n * (1 * 3 * 5 * ... * (2n-1))/(2^n * n!).]

Integrating term by term from 0 to x yields

sinh⁻¹ x = x + Σ(n=1 to ∞) C(-1/2, n) x^(2n+1)/(2n+1).

I hope this helps!

that's a typical sequence which you would be able to look up in tables of MacLaurin sequence. i might for this reason advise that "calculate" right here potential you may locate the sequence from undemanding suggestions. the final formulation for MacLaurin sequence is f(x) = f(0) + x*f '(0) + (x^2)/2*f "(0) + (x^3)/3! *f '''(0) + . . . so which you are going to be able to desire to distinguish quite a few cases and evaluate with x = 0 and placed the outcomes into this formulation.