( 538 ) 
The first nine terms contain of the power series for tan—! x only 
the first five coefficients. Together they give: tan—! 1 = 0.7821 
instead of 0.7854. 
Out of the first five terms of the power series we should find 0.8349, 
It is not difficult to transform the terms of the series of polynomials 
into a simpler form. 
After some deduction we get 
Eee) ita) 
or 
71 = ( le 1 zt 1 4 1 
Ge BN ky ee Age, NE: 
and by a similar notation it is evident that really 1 :)/ 2 is the mark 
of the series of polynomials. 
The example treated of here can serve to notice a phenomenon 
which will undoubtedly make its appearance in more intricate cases. 
In the series (VII) all kinds of values of the parameter a might 
have served to calculate tan—! 1. If for simplicity’s sake we consider 
only real values of a, then the following is to be noticed. 
For a = oo we should have obtained the narrowest region of con- 
vergence Gj, which encloses just the point «1; the mark of the 
series would have been 1. A wide region of convergence we should 
have had for instance for a= In the direction of the axis of 
Dor O9 
reality it would have extended to =) 5, and in 2 = 1 the mark 
of the series would have amounted to 3:4/13 = 0.83. The smallest 
mark 0.70 was obtained for a = 2, that is with a region of conver- 
gence reaching to a=// 3. 
Something similar will always take place as soon as we can 
enlarge or reduce the region of convergence by changing a para- 
meter. If by means of a series of polynomials we wish to find the 
value of F(x) in a given point 2, we must avoid an unnecessary 
extension of the region of convergence, but on the other hand we 
must not allow the boundary of this region to approach the point 
x too much. A region of convergence enclosing « not too narrowly 
and yet not too widely furnishes the best converging series of poly- 
nomials. 
