# Analytic Theory of Differential Equations by P. F. Hsieh, A. W. J. Stoddart

By P. F. Hsieh, A. W. J. Stoddart

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Thus its partial sums are: S" = " L m= O m � m! a a2 a" a2 a" S" = mI + I! + 2! + . . + n ! = I + a + "2 + . . + n! So = 1 . 3. Th e case o f the real exponential series is obtained a s a special case o f the complex series, if the point a is chosen on the real axis. The terms of the exponential sequence always converge to zero. The exponential se­ ries converges for every finite value of a . The convergence is so fast that the simulation window will only show a few of the 1000 calculated terms separately.

The sketch shows the first steps of the calculation for inscribed polygons with 2N corners, with N > 2. 3. Simulation. :ribed and circumscribed poly. gons. The simulation shows the appro�imations from the square to the polygon with 4096 comers. The square, with which the calculation starts, consists of 4 equal right triangles, whose cathetuses for the unit circle under consideration have length I . According to the theorem of Pythagoras, the hypotenuse of each triangle has the length h. The height h 4 is obtained via the theorem of Pythagoras using s4 /2 and the hypotenuse I of the lower triangle.

He uses the theorem of Pythagoras, the formula for the area of a right triangle, and symmetry considerations. From the above it follows that the baselines of the triangles constituting the polygons with n corners are given a� a simple function of n when doubling n. The following diagrams visualize the procedure. The first regular polygon, a yellow square, is circumscribed around the circle filled in gray; a second colorless square is inscribed in the circle. The inscribed polygon has a smaller area then the circumscribed one; The true value for the circle lies between the rwo.