I am trying to animate on maple. I have two solutions 𝑢1(𝑥, 𝑎) = 𝑐𝑜𝑠(𝑥 − 2𝑎) and 𝑢2(𝑥, 𝑎) = 𝑐𝑜𝑠(𝑥 + 2𝑎), for 𝑎 ∈ [1,10], how do I animate them?

S := Sum(1/i(i + 1)(i + 2)(i + 3), i = 1 .. infinity);
evalf(S);
                        -Float(infinity)
S1 := applyop(convert, [1, 2], S, parfrac, x);
                    infinity                      
                     -----                        
                      \                           
                       )              1           
              S1 :=   /     ----------------------
                     -----  i(i + 1)(i + 2)(i + 3)
                     i = 1                        
S2 := applyop(expand, 1, S1);
                    infinity                      
                     -----                        
                      \                           
                       )              1           
              S2 :=   /     ----------------------
                     -----  i(i + 1)(i + 2)(i + 3)
                     i = 1                        
S3 := map(op(0, S2), op(S2));
                                            /infinity    \
                                            | -----      |
                                            |  \         |
                                            |   )        |
                                            |  /     (-1)|
                                            | -----      |
                                            \ i = 1      /
            /infinity                      \              
            | -----                        |              
            |  \                           |              
            |   )                          |              
      S3 := |  /     i(i + 1)(i + 2)(i + 3)|              
            | -----                        |              
            \ i = 1                        /              
I am trying to find the limit approaching inf of the sum of 1/( 𝑖(𝑖 + 1)(𝑖 + 2)(𝑖 + 3)). I dont know what to do after this. 

lim 𝑛→+∞∑ 1 /(𝑖(𝑖 + 1)(𝑖 + 2)(𝑖 + 3))

Let 𝑦 and 𝑧 and be variables, let 𝑦 =( 𝑎+𝑏𝑧+𝑐𝑧^2)/( 𝑑+𝑒𝑧+𝑓𝑧 ^2) . Use the function solve in Maple to solve for 𝑧 in terms of 𝑦, that is, find a function 𝑧 = 𝑓(𝑦) (in fact, you should be able to find two). Let 𝑔(𝑧) = 𝑎+𝑏𝑧+𝑐𝑧^2)/( 𝑑+𝑒𝑧+𝑓𝑧 ^2)  and compute 𝑔(𝑓(𝑦)) for both of these functions and show that if you simplify the expressions you get a predictable answer. 

I ave already solved the values, however I am unable to simplify these expressions.

The Folium of Descartes (see the wikipedia page about this curve) is an algebraic curve defined by the equation 𝑥 ^3 + 𝑦 ^3 − 3𝑎𝑥𝑦 = 0, where 𝑎 is a constant. You will need the information on this wikipedia page in order to plot the following graphs. plot them for 𝑎 = 1,  Graph the curve with 𝑎 = 1 using the command implicitplot in the range of −3 ≤ 𝑥 ≤ 3 and −3 ≤ 𝑦 ≤ 3. (which i have already done)

However i am unable to Add a plot of the line 𝑥 + 𝑦 + 𝑎 = 0 on the same graph but dashed and with a different color (set numpoints as 10000).

In addition to defining a function in a standard way, a fast and convenient way to turn a formula (an expression) into a function is to use unapply command (check Maple help center for this command). The following items are about defining functions. 

1. Define a function midpoint, which returns the average of two arguments given on input. For example, midpoint(2,3) returns 2.5, midpoint(a,b) returns 𝑎+𝑏2. 
2. b) For two positive natural numbers 𝑎 and 𝑏 we can define the least common multiple lcm(𝑎,𝑏) via their greatest common divisor gcd (𝑎,𝑏), i.e.: lcm(𝑎,𝑏) = 𝑎𝑏gcd (𝑎,𝑏). When 𝑎 and/or 𝑏 are zero, lcm(𝑎; 𝑏)=0. Use the arrow operator and piecewise to define the function my_lcm which returns the least common multiple of two natural numbers. 
3. c) The quadratic formula allows to write the roots of 𝑝 = 𝑎𝑥2+𝑏𝑥+𝑐 explicitly in terms of 𝑎,𝑏, and 𝑐.