Power Series
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 1994 POWER RANGERS NEW SEASON TRADING CARDS BOTH SERIES US $17.99
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Power Series?
A fair coin is tossed repeatedly. The probability that the first head occurs on the nth toss is P(n)=(1/2)^n. When this action is repeated many times, the average # of tosses required until the first head is:
E(n)=sum from 1 to inf n*P(n)
This # is called the expected value of n
A) use the power series: 1/1-x = sum from 0 to inf of x^n, for |x|<1 to get a power series representation for f(x)=1/(1-x)^2 and give interval of convergence
B) what is the relation between E(n) and the series found in (a)?
C) what is the expected value of n? Be sure to give specific reason why you can make your conclusion.
I am completely clueless on this. If someone could just help me on each part not exactly answer it that would be very helpful. Thanks
A) 1/(1-x) = Sum(x^n, n=0 to inf) for |x| < 1
Take the derivative of both sides:
1/(1-x)^2 = Sum(n x^(n-1), n = 1 to inf) for |x|<1 [Taking the derviative give the same radius of of convergence.]
B) E(n) = Sum(n(1/2)^n, n=1 to inf) =
(1/2) Sum(n(1/2)^(n-1), n = 1 to inf) =
(1/2)(1/(1-(1/2))^2 [Plug x = 1/2 into summation above]
=(1/2)(1/(1/4)) = (1/2)*(4) = 2
C) The expected number of tosses to get heads is 2.
