Notation is a convention, a commonly shared interpretation of some symbols. Loops in programming languages can be written to decrease the index each time just as easily as they can increase it. The convention is to increase it, just like with Sigma and Pi notation, but they also support decreasing indeces. So, my opinion would be: sure! Why not? If I were to see an upper index value that is smaller than the lower one, my first assumption would be that I would need to decrease the index by 1 for each iteration — which seems to be what you intend.
After expanding the Pi notation into the full expression that it represents, the person working with that expression must follow the rules of algebra or matrix algebra , and the index number of each factor would not have any effect on such rules. But, perhaps I do not understand the situation you seek to describe.
So will provide the correct sign for the nth term. Putting the three thoughts above together, I get:. Furthermore is there a way of simplifying the notation and finding a result that is a function of n? If I have interpreted the expression you show correctly, it is neither an arithmetic nor a geometric sequence. As n grows, the constant power of 2 in the expression will dominate the initial results a lot more, but the infinite number of subtractions from it will eventually catch up to its value, no matter how large it is.
Thanks for your clear explanations. It helps me to understands the notation means and how to use it. Using Pi notation, I interpret your question to be. Terms with odd values of i will be negative. Hello, I am trying to utilize the Pi notation to represent a repeating multiplication, but one that rounds up to the nearest whole after each time there is a multiplication or division.
Before I continue please forgive my mathematical illiteracy, I am taking an amateur interest in this. What I am wondering about is this. In my mind, this rounds up each time the value is divided by 1-r. The notation that follows a capital Pi describes only the term that is to be multiplied.
The difficulty you describe is that you wish to specify what happens to the result of that product, and capital Pi notation does not provide any means to do that. Two ways to resolve the problem come to mind: 1 your expansion of the problem using square brackets 2 using a programming language to describe a loop in which each product is then rounded, before repeating the loop until the specified number of multiplications have been carried out. How should I proceed if I want to get it for n instead of 3.
Equation for Xn in terms of P1,P2,……Pn. Summation notation does not provide an easy way that I can think of to do what you describe. While it can add a bunch of terms very nicely, the challenge is describing each of the terms you show as a function of the term number.
This would be easy to do in a computer program, but not so much using summation notation. This problem is not strictly a Pi Notation problem, as it involves a limit and a power outside of any Pi Notation. Also, I am not certain where the product you describe is supposed to end. However, I have never worked with infinite products.
Your answer options suggest that there is some expansion of a a logarithm that results in an infinite product of tangent functions, however I am not familiar with that. If each factor described by the pi notation contains an instance of a the variable, you would need to use the product rule… potentially many times. However, if each factor does not contain the variable or a function of the variable that you are differentiating with respect to, then the whole product would be a constant.
So, depending on the number of factors in the product, it could be a very long process, or a very short one. I have a question, please. Is there a way to rewrite the following expression using both sigma and pi?
I think this would do it. Pi is not needed:. Sir, this is a very helpful website. This formula reflects the commutative property of infinite double sums by the quadrant. It takes place under restrictions like , which provide absolute convergence of this double series.
This formula shows how to change the order in a double sum. This formula reflects summation over the infinite triangle in a different order. This formula reflects summation over the infinite trapezium in a different order. This formula reflects summation over the infinite trapezium quadrangle in a different order. This formula shows the summation over the infinite trapezium quadrangle in a different order.
This formula shows the summation over the trapezium quadrangle in a different order. This formula shows summation over the trapezium quadrangle in a different order. This formula shows how to change the order of summation in a triple sum. Many thanks. Show 3 more comments. Active Oldest Votes. This is easy to see. Add a comment. Sign up or log in Sign up using Google.
Sign up using Facebook. Sign up using Email and Password. Each addend in the sum will be the square of an index value. The index values begin with 3 and increase by 1 until reaching 7. Thus, we have the index values 3, 4, 5, 6, and 7, and the squares of those are 9, 16, 25, 36, and In some cases we may not identify the upper limit of summation with a specific value, instead usingf a variable. Here's an example.
The lower limit of summation is 0 and the upper limit is n. Each addend in the sum is found by multiplying the index value by 3 and then adding 1 to that. Because we do not know the specific value for n, we use an elipsis.
0コメント