Major Error on Calculator involving Series

[jte]

(02-29-2016 04:06 AM)Han Wrote:  …
In fact, even with the most powerful hardware, calculators are limited by... well, their hardware. For example, when one graphs \( sin(ax) \) where \( a \) is a large value, the graph (on any calculator or even computer) will be wrong for sufficiently large \( a \) simply because the pixel density will eventually be insufficient if the frequency in the graph is high enough that several peaks or valleys of the sinusoidal graph occur within the width of one single pixel. And graphing is one of the most basic features of graphing calculators. I am of the opinion that this bug will never be completely eradicated since screens will always be physically limited by the number of pixels it can display.
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Han, you might want to try the Advanced Graphing app and look at
http://www.dgp.toronto.edu/~mooncake/pap...Tupper.pdf

A problem with most automated graphing techniques is not they can be wrong, but that they cannot be right as there is no precise definition of what their output means. (Or is supposed to mean: what can one infer of the mathematical relation, given the image shown by the device?) A strict interpretation that remains understandable to those not versed in the intricate details of the construction and limitations of the typical techniques employed is that: this is the output of the algorithm. (Which isn't very revealing…) There are techniques that allow for some information to be revealed by a computer-generated image. (E.g.,: this pixel is black -> there is a point in the Cartesian region the point represents that belongs to the relation being graphed. An implementation that could satisfy such requirements, if it uses floating-point, must keep track of things like round-off and properties of the function(s) being evaluated.)

The Advanced Graphing app takes some steps towards addressing the shortcomings you mention. Some additional steps remain to be taken.