perryben said:w[n] = exp(j*pi*n)
w[n] = (−1)^n
Hi, can anyone off hand verify why these two exressions are equal? Thanks
To verify w[n] = (-1)^n means to prove that the function w[n] alternates between positive and negative values, with each value being the opposite sign of the previous value. In other words, when n is an even number, w[n] will be positive, and when n is an odd number, w[n] will be negative.
To verify w[n] = (-1)^n, you can use mathematical induction. This method involves proving that the statement holds true for the base case (usually n = 0 or n = 1) and then assuming it holds true for n = k and proving it for n = k+1. If the statement holds true for the base case and can be proven for any arbitrary k, then it can be said to hold true for all n.
Verifying w[n] = (-1)^n is important because it is a fundamental property of alternating functions. It also has many applications in mathematics and science, such as in signal processing and in solving differential equations.
Yes, w[n] = (-1)^n can also be verified using other methods such as direct proof and proof by contradiction. However, mathematical induction is the most commonly used method for verifying this type of alternating function.
Yes, there are many other functions that alternate between positive and negative values, such as sin(x), cos(x), and tangent(x). These functions are known as periodic functions and have a repeating pattern of positive and negative values.