Series and transform1. Evaluate the sum of the following series: ¦ ¸ ¹ · ¨ © 999 § 1 10 1 log n n n [Hint: write out first 3 or 4 terms and last 3 or 4 terms] [6]
2. Show that the first four terms of the Maclaurin series expansion of y(x), where loge (y) = e x , are as follows: ¸ ¸ ¹ · ¨ ¨ © § ... 6 5 1 3 2 x y e x x [10] Calculate the percentage error between the true value of y and that derived from the first four terms of the above expansion when x = 0.3 .
3. A voltage v(t) is applied to a coil to switch on a magnetic field periodically for an interval of S seconds. The voltage as a function of time t is described by the function below: v(t) = 4 for 0 < t < S/2 v(t) = í4 for -S < t < v(t) = 0 for -S < t < -S and S < t < S a) Sketch v(t) over the interval from t = -S to t = +S. If this function v(t) is assumed to be periodic over an interval of 2S, it is possible to express the function as a Fourier series as follows: + ( a mt+ b mt) 2 a v t = m m m=1 0 ( ) ¦ cos sin f .
b) Use the Fourier series integrals to show that: am 0 for all values of m (including m = 0), and that: > @ 1 cos( / 2) 8 S S m m bm . [20] c) Thus show that v(t) can be represented by the following series: ¸ ¹ · ¨ © § sin7 ... 7 1 sin6 3 1 sin5 5 1 sin3 3 1 sin sin 2 8 v(t) t t t t t t S .
d) Use Matlab to plot the above series expansion of v(t) over the interval from t = -S to t = +S using only the first two, three, four, five, six, and one hundred terms (i.e. plot six graphs on the same axes). Briefly describe (in words) the effect of including an increasing number of terms in the series.