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Chris Fox's Engineering Section


Sequences and Series

Consider a series whose terms are a, ar, ar², ar³ etc. This is an example of geometric progression, and r is termed the common ratio. If a=1 and r=0.5, for example, the sequence becomes 1, 0.5, 0.25, 0.125 etc, and converges to 0.

If the geometric series is summed, ie we write a+ar+ar²+ar³+...+arn-1, the sum Sn=a{(rn-1)/(r-1)}. If |r|<1, the sum to infinity S=a/(1-r). |r| denotes the absolute value of r, which means the value without regard to whether r is positive or negative.

In general, a geometric series will converge to a definite sum if |r|<R, where R is the radius of convergence.

Consider the series S=1+2x+3x²+4x³ → ∞. Multiplying both sides by x gives xS=x+2x²+3x³+4x4 → ∞. Subtracting the second equation from the first gives S-xS=1-x+2x-2x²+3x²-3x³+4x³-4x4..., and simplifying gives S(1-x)=1+x+x²+x³.... If |x|<1, then S(1-x)=1/(1-x), and S=1/(1-x)².

Trigonometric functions can be represented by series. For example, cos x = 1-(x²/2)+(x4/24).... This works for any value of x.

A series can be summarised using sigma notation. The maximum and minimum values of a variable are placed respectively above and below the capital sigma, with the formula following it. For example, 1+2+3+...+N may be summarised as Σn=1N n. This, incidentally, is an example of an arithmetic series, and its sum is N(N+1)/2, or (N²+N)/2.


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