Things You Need to Know

for Probability & Statistics

1. 1+a+a²+a³+a⁴+...+aⁿ  = (a^n-1-1)/(a-1)  if a1 and n is a natural number.
         So, for example, 1+3+27+81+243 = (3⁵-1)/(3-1)  using a=3 and n=4.
2. 1+2+3+...+n = n(n+1)/2 for all natural numbers n.
         So, for example, 1+2+3+4+5+6 = 6·7/2 = 21  using n=6.

1. 1+a+a²+a³+a⁴+...+aⁿ+...  (an infinite series, unlike the finite series above)
             = 1/(1-a)  if |a|<1 .
         So, for example, 1+1/2+1/4+1/8+... =1/(1-1/2)  using a=1/2.

The following three are logically equivalent, for real numbers x,,  and >0:

• x(- , +)
• - < x < +
• |x-|<

f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + f⁽⁴⁾(0)x⁴/4! + ...+ f⁽ⁿ⁾(0)xⁿ/n! + ...
which can be written as
  f⁽ⁿ⁾(0)xⁿ/n!
For nearly all the functions you will meet in lower-level courses, this series converges to f(x) on some interval centered at 0.