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If you re baffled by the idea that we can add up infinitely many numbers and get a finite sum, you can use the frog-and-wall analogy Imagine that a frog sits 8 meters (8 m) away from a wall Then she jumps halfway to the wall, so she s 4 m away from it Now imagine that she continues to make repeated jumps toward the wall, each time getting halfway there (Fig 19-3) No finite number of jumps will allow the frog to reach the wall To accomplish that goal, she would have to take infinitely many jumps This scenario can be based on a sequence of partial sums of a series S = 4 + 2 + 1 + 1/2 + 1/4 + 1/8 + A real-world frog cannot reach the wall by jumping halfway to it, over and over But in the imagination, she can There are two ways this can happen First, in the universe of mathematics, we have an infinite amount of time, so an infinite number of jumps can take place Another way around the problem is to keep halving the length of time in between jumps, say from 4 seconds to 2 seconds, then to 1 second, then to 1/2 second, and so on This will make it possible for our cosmic superfrog to hop an infinite number of times in a finite span of time Either way, when she has finished her journey and her nose touches the wall, she ll have traveled exactly 8 m Therefore, the sum total of the lengths of her jumps is S = 4 + 2 + 1 + 1/2 + 1/4 + 1/8 + = 8

Here s a challenge!

Section A: Initial change request Project name Request submitted by Request submitted by Purpose of change Brief description of change List of documents and components for which change is requested Value of change: Does requester think it is essential If not, what value is assigned Name of project (add sub-projects a ected, if appropriate) Name of person making request Department, user group, or team, and contact information Simple statement of why the change is either necessary or valuable What change will be made What existing documents, such as speci cations, or components, such as prototype screens, or reports given to the user for review need to be changed Include all items.

Consider the series that we dealt with in A more sophisticated example a couple of paragraphs ago, but only up to the reciprocal of the nth power of 2 Let Sn be the partial sum of this series up to, and including, that term Write Sn in summation notation

The product g,g, is the open-loop vector. If the inner loop has been adjusted for f/4-amplitude damping, its open-loop gain will be 0.5 at the period of oscillation. But the phase lag at the period of oscillation is 180 , which makes the gain vector 0.5, L-180 , or -0.5. The closedloop vector go2 at the natural period is then g02 = -0.5 = -1.0 1.0 - 0.5 or

We can simplify this problem by remembering a few basic facts in trigonometry, and by applying a little algebra First, let s remember that the cosecant is equal to the reciprocal of the sine, so the converse is also true We have 1/(csc q) = sin q When we square both sides, we get 1/(csc2 q) = sin2 q Substituting in the equation for our function gives us f (q) = (sec2 q) (sin2 q) We ve learned that the secant is equal to the reciprocal of the cosine We have sec q = 1/(cos q) so we can square both sides to get sec2 q = 1/(cos2 q)

Substituting again in the equation for our original function, we obtain f (q) = (sin2 q)/(cos2 q) = [(sin q)/(cos q)]2 The sine over the cosine is equal to the tangent, so we can substitute again to conclude that our original function is f (q) = tan2 q with the restriction that we can t define it for any input value where either the secant or the cosecant become singular The solid black curves in Fig 15-10 show the result of squaring all the values of the tangent function, noting the additional undefined values as open circles At the points shown by the open circles, the cosecant function is singular so its square is undefined That means we can t define our ratio function f at any such point At the asymptotes (dashed vertical lines), the secant function is singular so its square is undefined, making it impossible to define the ratio function f for those values of q Our function f has a period of p The domain of f includes all real numbers except integer multiples of p /2, where one or the other of the original squared functions is singular The range is the set of all positive reals

Be as speci c as possible The project team will verify, and possibly add to, this section Pick one: Essential (core): product or service will fail in production, or project will not be delivered at all, if change is not made Not essential: de ne the value of the change, so it can be compared to cost and risk factors This could be measured on a simple high/medium/low scale, or it could be a dollar value calculated as change in ROI If the value of the change is not yet agreed upon, state the disagreement and identify the issues What happened that makes this change desirable Select from the list of sources of change below.

f (q )