In this problem you will prove the geometric series summation formula, which was
ID: 3143720 • Letter: I
Question
In this problem you will prove the geometric series summation formula, which was illustrated in class by the example 1/2 + 1/4 + 1/8 + .. = 1. (a) Let x be a real number, and define the sum S: = x + x^2 + x^3 + .. Now write down x middot S and show that xS = S - x. Solving for S, you should find that S = x/1 - x. Plug in x = 1/2. (b) Think carefully about what happens if you set x = 2. Does the formula make sense? Were there any implicit assumptions in the argument from part (a) that are causing problems? (c) Now consider a finite geometric summation: again, for real x and some natural number n, let S_n: = x + x^2 + x^3 + .. + x^n. Use a similar argument to part (a) and derive a formula for S_n. Does your answer make sense for all x?Explanation / Answer
(a) S = x + x2 + x3 + ...... (1)
=> S = x (1 + x + x2 + x3 + ......) (2)
Substituting (1) in (2)
=> S = x (1 + S)
=> S = x + xS
=> S - xS = x
=> S (1 - x) = x
=> S = x / (1 - x)
When x = 1/2,
S = 1/2 / (1 - 1/2)
=> S = 1/2 / 1/2
=> S = 1
(b) When x = 2,
S = 2 / (1 - 2)
=> S = 2 / -1
=> S = -2
Thus we arrive at a negative value which is absurd. The reason is because when x = 2, the series is diverging and not converging.
The essential condition is x should be less than 1.
(c) Sn = x + x2 + x3 + ..... xn
=> Sn = x (1 + x + x2 + ..... xn-1)
Note that 1 - xn = (1 - x) (1 + x + x2 + ..... xn-1)
=> Sn = x (1 - xn) / (1 - x)
The above formula works for all x except for x = 1 where it is undefined.
In the case of x = 1, Sn is simply n.
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