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Explorations of d'Alembert's Series

The command SquareRoot[x,n] computes the first  n terms of the binomial series expansion of a and compares it to the actual value (10-digit accuracy).

> SquareRoot := (x,n) -> [1+evalf(sum((-1)^(k-1)*(2*k-2)!/2^(2*k-1)/k!/(k-1)!*x^k,k=1..n),10),evalf(sqrt(1+x),10)];

> SquareRoot(200/199,100);

The next command computes  this function at ten values of n: m/10, 2m/10, ..., m. If we chooose a value of m that is not a multiple of 10, it computes these at the floors. This is useful so that the number of terms is not always even or always odd.

> SquareRootList := (x,m) -> [seq([floor(j*m/10),SquareRoot(x,floor(j*m/10))[1]],j=1..10)];

> SquareRootList(200/199,1007);

The last command plots these values.

> plots[listplot](SquareRootList(200/199,1007));