数学函数代写 MTH1030/35代写 Sequences代写 Serious series代写

MTH1030/35: Assignment 3

数学函数代写 Step by step to infinity (Note that this due date is a Tuesday not a Thursday like the previous two assignments!) The Rules of the Game

Step by step to infinity

(Note that this due date is a Tuesday not a Thursday like the previous two assignments!)

The Rules of the Game

Before starting to work on this assignment, please make sure that you understand the instructions on Moodle that go with our assignments (Assignment 3 (=Assignment 1) rules and F.A.Q. plus Academic integrity, plagiarism, and collusion policy).

Your submission for this assignment should be ONE pdf file.

Always justify your answers.

This assignment is worth 100 marks.

Now, let’s have some fun!

1 Sequences 数学函数代写

Find an example of a function f such that the corresponding sequence converges and such that f(x) = x has exactly 1 solution.

Find an example of a function f such that the corresponding sequence converges and such that f(x) = x has more than one solution.

Find an example of a function f such that the corresponding sequence diverges and such that f(x) = x has no solution.

Find an example of a function f such that the corresponding sequence diverges and such that f(x) = x has a solution.

Hint: This and some of the other questions in this assignment ask you to come up with some examples of functions and series that have certain properties. ALL of these questions have VERY simple functions and series as answers. Please don’t overcomplicate things 🙂

2 Serious series 数学函数代写

2.1 [10 marks]

The alternating series test is a theorem which says the following: If a series

a₁ − a₂ + a₃ − a₄ + a₅ − …

satisfies (1) a_n > 0, (2) a_n ≥ a_n+1 and (3) lim_n→∞ a_n = 0, then the series converges. Very powerful and very useful. For example, the series

satisfies (1), (2), and (3).

a) Give an example of a divergent series that satisfies (1) + (3) but not (2).

b) Give an example of a divergent series that satisfies (1) + (2) but not (3).

 

2.5 [10 marks]

Consider the unit square in the xy-plane whose corners are (0, 0), (1, 0), (0, 1) and (1, 1). Subdivide it into nine equal smaller squares and remove the square in the centre. Next, subdivide each of the remaining eight squares into nine even smaller squares, and remove each of the centre squares. And so on. The following diagram shows what’s left after the

first three steps of this construction. Give an example of a point in the original square that never gets removed. Show that the area of what is left over when all those squares have been removed is 0, by verifying that the sum of the areas of all the removed squares is 1.

3 A cat and mouse dog game [25 marks] 数学函数代写

Let’s play a game. I am thinking of two differentiable functions cat(x) and dog(x). Both functions can be written as power series

(a) Find cat(x) and dog(x) by calculating the general terms c_n and d_n of their power series.

Hint: To figure out what the coefficients of the two power series are use the fact that two power series are equal if and only if corresponding coefficients are equal and note that cat’’(x) = cat(x). [10 marks]

(b) Write the functions e^x and e^−x in terms of of cat(x) and dog(x). [5 marks]

(c) Conversely, express cat(x) and dog(x) as a combination of e^x and e^−x. [5 marks]

(d) Using Mathematica or another piece of software plot cat(x) and the first four different partial sums of its power series in the interval [−2, 2].¹ [5 marks]

¹ In Mathematica several functions can be plotted in the same diagram as follows: Plot[{Cos[x], Cos[2 x], Cos[3 x]}, {x, 0, 2 Pi}, PlotLegends -> “Expressions”]

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