One can compute an , where a > 0 and n is a positive integer in a brute-force way by repeatedly multiplying a: an = a
9x", where n is a positive integer. For what values of n is 0 The number 0 is a critical point of the autonomous differential equation dx/dt asymptotically stable? Semi- stable? Unstable? Asymptotically stable: Onis odd Onz9 Ons o nis even Semi-stable: n s 0 Onis even n is odd On<9 Unstable: n s 0 n is even Onis odd Ons 0 Repeat for the differential equation dx/dt =-9xn Asymptotically stable: On0 On is odd nis even Onso Semi-stable: On...
Where n is any positive integer, do the following: A. For ε > 0, prove that an converges to a limit of 4 by using the formal definition of convergence of a sequence to a limit, showing all work. 1. Justify each step as part of your proof in A.
Consider the probability distribution ?(?) = ??n, 0 ≤ ? ≤ 1 for a positive integer ?. Derive an expression for the constant ?, to normalize ?(?). Compute the average 〈?〉 as a function of ?. Compute the expectation value of the second moment. Compute the variance as a function of ?.
1. We can determine how many digits a positive integer has by repeatedly dividing by 10 (without keeping the remainder) until the number is less than 10, consisting of only 1 digit. We add 1 to this value for each time we divided by 10. a) Describe the algorithm in a recursive way b) Implement this recursive algorithm in C++ and test it using a main function that calls this with the values of 15, 105 and 15105. (Hint: Remember...
Question 10. Consider the function defined by f(n) = 2n where n is a positive integer. (i) Can this function be computed by a Turing machine? Why or why not? ( ii) Is this function primitive recursive? Why or why not?
For any positive integer n, prove that .Hint for one way of proving this: If X is set of cardinality n, find a bijection between the set of o i is even i is odd elements of P(X) of even cardinality and the set of elements of P(X) of odd cardinality.
A 3-digit positive integer N is randomly chosen. Compute the probability of the event that (a) N is divisible by 3. (b) N is divisible by 3 if its leftmost digit is 1.
7. Determine the DTFT of the sequence [n]-, where N is 1S 0 otherwise a positive integer. Plot it for No 4 and No 20
Prove by induction, where n is a positive integer that 2 2*3 n(n+1) n+1