The exponential function can be defined by the power series as: 80 sin(z) = Σ (-1)* k=0 (2k+1)! Z z5 2k+1 Z- + 3! 5! - 7! +... a) Using recursive function technique, write a function sinusoidal inputs: z and k. The function will output the corresponding value. your recursive function with z = 2 and k=3 >> sinusoidal (2,3) (z, k) that takes 2 In your script, test ans == 0.9079 b) Consider that z=1. Write a script that will loop through values of k until the difference between the series and the actual value (which is sin(1)) is less than 0.0001. The script should then print out the built-in value of sin(1) and the series approximation to five decimal places, and also print the value of k required for such accuracy.

Write a recursive function for int powerOfTwo (int k). The function determines the value of 2k. (Note: k is a positive integer). Example, when k=0 the function returns 1 and when k-3 the function returns 8. To show that your code is correct, give the recursive trace for powerOfTwo (2) that returns 4.

Write recursive functions for the following equations:a. Harmonic number is defined by the equationb. Fibonacci numbers are defined by the formula:FN = FN-2 + FN-1, for N ≥ 2 with F0 = 0 and F1 = 1

Write a recursive findodd function which returns the number of odd digits of given integer. Write a C program that tests function findOdd. (Your function must be recursive.) (Examples: findodd(22314675) returns 4, findOdd(113356) returns 5) Sample Input1: Sample Output1: 123312 Sample Input2: Sample Output2: 8. 11335573 Sample Input3: 1143 Sample Output3:

In programming, a recursive function calls itself. The classical example is factorial(n), which can be defined recursively as n*factorial(n-1). Nonethessless, it is important to take note that a recursive function should have a terminating condition (or base case), in the case of factorial, factorial(0)=1. Hence, the full definition is: factorial(n) = 1, for n = 0 factorial(n) = n * factorial(n-1), for all n > 1 For example, suppose n = 5: // Recursive call factorial(5) = 5 * factorial(4) factorial(4) = 4 * factorial(3) factorial(3) = 3 * factorial(2) factorial(2) = 2 * factorial(1) factorial(1) = 1 * factorial(0) factorial(0) = 1  // Base case // Unwinding factorial(1) = 1 * 1 = 1 factorial(2) = 2 * 1 = 2 factorial(3) = 3 * 2 = 6 factorial(4) = 4 * 6 = 24 factorial(5) = 5 * 24 = 120 (DONE)   Exercise (Factorial) (Recursive): Write a recursive method called factorial() to compute the factorial of the given integer. public static int factorial(int n) The recursive algorithm is:…

Write a recursive function that takes an array and a callback function and returns True if any value of that array returns True from that callback function otherwise returns False.As a callback funtion for this problem, assume that the callback function is used to check if a given input is an even number.- Analyse the running time of the algorithm

Write a recursive function that returns true if the digits of a positive integer are in increasing order; otherwise, the function returns false. Also, write a program to test your function.

Write a program using recursive function that calculates the sum of the first no elements from harmonic series and print the main function. Also, the function displays the series elements. n is entered by the user. The nth harmonic sequence element is calculated as 1/(a + (n-1)d). Consider a=1, d=1, and n >0

Write a recursive function to compute the following series: m(i) = 1 + 1/2 + 1/3 + . . . + 1/i Write a test program that displays m(i) for i 1, 2, ..., 10.

In Lisp Programming (Please show that it does work) Write a simple recursive function to compute the f(n) for the following sequence: f(n) = f(n-1) + f(n-3) f(0) = 0 f(1) = f(2) = 1 Optimize this function using dynamic programming. You can assume that the number n in the function call for testing will be less than 20.