Exercise 0.1
a)

s=Sum[2n+1,{n,0,19}]

p=Product[2n+1,{n,0,19}]

b)

s/p

p^s

c)

s^p

(There would be many billions of pages of result, and
 time spent accordingly)

d)

Clear[s,p]

Exercise 0.2

D[1/(x^2+1)^2,x]

Integrate[1/(x^2+1)^2,x]

Exercise 0.3

Plot[1/(x^2+1)^2,{x,-5,5}]

Exercise 0.4

For [n=0,
     n!<=10^n,
     n=n+1,
     Print[n," ",n!," ",10^n]
    ]

Print[n," ",n!," ",10^n]

Exercise 0.5

Do[Print[n," ",Gamma[n]," ",(n-1)!],{n,1,50}]

Exercise 0.6
a)

fact=20!

b)

time=(10^9*365+10^9/4)*24*60*60

c)

fact/time

d)

fact=.

Clear[time]

Exercise 0.7

Function[x,x^3-3x+1124][{-1234,-123,-12,0,12,123,1234}]

Exercise 0.8

g[x_]:=(x^3-x+2)/(x^5-3x+27)

Plot[g[x],{x,-10,10}]

Horizontal asymptote is y=0. For vertical asymptotes:

NSolve[x^5-3x+27==0,x]

For local minima and maxima:

NSolve[D[g[x],x]==0,x]

Exercise 0.9

Plot3D[4Sin[4x]+10Cos[y],{x,-Pi,Pi},{y,-Pi,Pi}]

Exercise 0.10

Unprotect[Sum]

Sum=Product

Sum[i,{i,0,50}]

As expected, we got the product.

Clear[Sum]

Now Sum got its default functionality back:

Sum[i,{i,0,50}]