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Phasor Mathematics
13 questions By Tony R. Kuphaldt
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Question 4 of 13
Euler’s constant, that ubiquitous constant in mathematics symbolized by the letter e, may be found as the result of the following mathematical series:
$$e = \sum_{n=0}^\infty \frac{1}{n!}$$
Approximate the value of e in steps, using the following table:
n n! 1 / n! ≈ e 0 1 2 3 4 5 6 7
Also, express this series as a partial sum up to n = 7.Reveal answern n! 1 / n! ≈ e 0 1 1 1 1 1 1 2 2 2 0.5 2.5 3 6 0.16667 2.66667 4 24 0.04167 2.70833 5 120 0.00833 2.71667 6 720 0.00139 2.71806 7 5040 0.00020 2.71825
Shown here is the partial sum up to n=7:1 0!+ 1 1!+ 1 2!+ 1 3!+ 1 4!+ 1 5!+ 1 6!+ 1 7!. . . and again, in slightly different form . . .
1 + 1 + 1 2+ 1 6+ 1 24+ 1 120+ 1 720+ 1 5040Notes:It should go without saying that students should consult their electronic calculators (or a textbook) to see what the actual value of e is, and compare that to their partial sum approximation.
This questions provides students with the opportunity to see where e comes from, and also to explore the behavior of a series by calculating partial sums. Many students find it fascinating to see the sequence of partial sums converge on the true value of e, seeing how this formerly mysterious constant can actually be calculated using nothing more than repeated arithmetic.
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Question 5 of 13
The trigonometric function of cosine may be found as the result of an infinite series. Note that this series assumes the angle x to be expressed in units of radians, not degrees:
$$cos \ x = \sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!}$$
Approximate the cosine of 1 radian (cos1) in steps, using the following table, then write the partial sum expansion up to n = 5:
n (−1)n x2n (2n)! ≈ cosx 0 1 2 3 4 5 Reveal answern (−1)n x2n (2n)! ≈ cosx 0 1 1 1 1 1 -1 1 2 0.5 2 1 1 24 0.5416667 3 -1 1 720 0.5402778 4 1 1 40320 0.5403026 5 -1 1 3628800 0.5403023
Shown here is the partial sum expansion up to n = 5:$$cos \ x \approx 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \frac{x^8}{8!} - \frac{x^{10}}{10!}$$
Notes:It should go without saying that students should consult their electronic calculators to see what the actual value of cos1 is, and compare that to their partial sum approximation. Students should also feel free to explore the validity of this series by approximating the cosine of angles other than 1 radian.
This questions provides students with the opportunity to see how cosx may be arithmetically calculated. This, in fact, is how many digital electronic computers determine trigonometric functions: from partial sum approximations.
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Question 6 of 13
The trigonometric function of sine may be found as the result of an infinite series. Note that this series assumes the angle x to be expressed in units of radians, not degrees:
$$sin \ x = \sum_{n=0}^\infty (-1)^n \ \frac{x^{2n+1}}{(2n + 1)!}$$
Approximate the sine of 1 radian (sin1) in steps, using the following table, then write the partial sum expansion up to n = 5:
n (−1)n x2n 1 (2n 1)! ≈ sinx 0 1 2 3 4 5 Reveal answern (−1)n x2n 1 (2n 1)! ≈ sinx 0 1 1 1 1 1 -1 1 6 0.8333333 2 1 1 120 0.8416667 3 -1 1 5040 0.8414683 4 1 1 362880 0.8414710 5 -1 1 39916800 0.8414710
Shown here is the partial sum expansion up to n = 5:$$sin \ x \approx x - \frac {x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \frac {x^{11}}{11!}$$
Challenge question: which series (sine or cosine) would a computer be able to calculate quickest, given a certain number of digits (precision) past the decimal point? In other words, which of these two infinite series converges fastest?
Shown here is a partial sum expansion of the cosine function, for comparison:
$$sin \ x \approx 1 - \frac {x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \frac {x^{10}}{10!}$$
Notes:It should go without saying that students should consult their electronic calculators to see what the actual value of sin1 is, and compare that to their partial sum approximation. Students should also feel free to explore the validity of this series by approximating the sine of angles other than 1 radian.
This questions provides students with the opportunity to see how sinx may be arithmetically calculated. This, in fact, is how many digital electronic computers determine trigonometric functions: from partial sum approximations.
