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lim xn 1 0 for x < 1
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we nd that eq. (4-A) becomes
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lim Sn
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1 for x < 1 1 x
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Appendix
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Hence, if the number of terms n is allowed to increase without bound, Sn becomes equal to 1 and thus eq. (3-A) can be written as 1 x 1 1 x x2 x3 xn for x < 1 5-A 1 x which, it should be understood, is EXACTLY TRUE only in the limit as n becomes in nitely great. In the same way, the functions x , sin x, and cos x can be represented by power series in x. In these cases, however, the nature of the series is such that the series representation is valid for ALL positive and negative values of the variable x.
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Series RL Circuit. L/R Time Constant
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We must rst note the nature of the negative exponential function,  x 1=x , where  (epsilon) denotes the irrational number  2:71828 . . . ; de ned by eq. 146 in section 6.5. In the discussion here, we ll be interested only in the case where x is a positive real number. Using your calculator, you can verify the values listed in the following table of values (values of  x rounded o to two decimal places). These values are plotted against x in Fig. 18-A.
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x 0.00 0.10 0.20 0.30 0.40 0.50 0.70
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 x 1.00 0.91 0.82 0.74 0.67 0.61 0.50
x 0.80 0.90 1.00 1.50 2.00 3.00 5.00
 x 0.45 0.41 0.37 0.22 0.14 0.05 0.01
Fig. 18-A
Now consider the basic series RL circuit, to which a constant voltage of V volts is applied at the closing of a switch, as shown in Fig. 19-A. In Fig. 19-A, L is inductance in henrys, R is resistance in ohms, and i is current in amperes owing any time t seconds after the switch is closed at t 0. At t 0 the current i is zero, at which time the entire applied voltage V appears across the coil L; then, as time increases, the current increases slowly toward the limiting value of I V=R, as shown in Fig. 20-A. As this occurs, the voltage drop across L decreases, while the voltage drop across R rises toward the limiting value of IR V volts. The exact relationship between the current i and time t is given by the equation i V 1  Rt=L R 6-A
Note that when t 0, then i 0, as already mentioned, and as shown in Fig. 20-A. Then, as time increases, the term  Rt=L decreases exponentially toward the value zero (as
Appendix
Fig. 19-A
Fig. 20-A
in Fig. 18-A). Thus, as time t increases, the current i increases toward the limiting value of i I V=R amperes, as shown in Fig. 20-A. In Fig. 20-A, note that time is expressed in multiples of L/R. This can be done because the ratio of henrys to ohms is time in seconds, as the following shows. By eq. (181), L and, by Ohm s law, 1 amp R volts hence, L volts sec amp seconds R amp volts The ratio of henrys to ohms, L/R, is called the time constant of the basic series circuit of Fig. 19-A. As Fig. 20-A shows, at the end of one time constant (L/R seconds) the current in Fig. 19-A will have risen to approximately 63% of its nal value of I V=R amperes. (To show this, set t L=R in eq. (6-A).) v volts volts sec di=dt amp=sec amp
Note 14.
Series RC Circuit. RC Time Constant
Here we wish to emphasize that time is required to change the amount of energy stored in the electric eld of a capacitor. To illustrate this, consider the basic series RC circuit, to which a constant voltage of V volts is applied at the closing of a switch, as shown in Fig. 21-A. We wish to examine the manner in which the VOLTAGE ACROSS THE CAPACITOR increases after the switch is closed. In Fig. 21-A, R is resistance in ohms, C is capacitance in farads, and i is current in amperes owing at any time t seconds after the switch is closed. We ll assume that initially (at t 0) there is zero voltage across the capacitor. We rst note that, at the instant the switch is closed at t 0, the capacitor momentarily behaves like a short circuit ; thus, at t 0 the current is equal to V/R amperes. Then, as time increases and the capacitor begins to charge, the current i decreases exponentially, in
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