Series and Expansion Formulas you Must Know

 

In a Nut Shell:  You need to remember the series expansions for exponential and trig functions as they

come in handy when you manipulate series expansions, evaluate derivatives  and integrals of functions.

 

 

Maclaurin series expansions for  ex, sin x, and cos x are as follows:

 

  ex       =   1 + x + x2/2! + x3/3! + .  .  .  + xn/n!

                                                                                              ∞

 sin x   =   x – x3/3! + x5/5! + .  .  .  +(-1)n x2n+1/(2n+1)!  =  ∑  (-1)n  x2n+1/(2n+1)! 

                                                                                            n=0

 

                                                                                        ∞     

cos x   =   1 – x2/2! + x4/4! + .  .  .  +(-1)n x2n/(2n)!   =   ∑  (-1)n  x2n /(2n)!

                                                                                      n=0

 

1/(1 + x)  =  1 – x + x2 - x3 + ….  +(-1)n xn    (by direct division)     n  =  0, 1, 2, ..

 

The binomial series is:                           Click here for an example.

 

( 1 + x )a   =   1 + ax + a(a-1) x2/2! + a(a-1)(a-2) x3/3!  +  . . . 

 

                        ∞     

( 1 + x )a     =    ∑  { a/n } xn     where    { a/n } =  [ a(a˗1)(a˗2)  .  .  .  .  .  (a ˗ n+1 ) ] / n!

                       n=0     

 

The Taylor series is:  (If  a  =  0,  then this series is called the Maclaurin series.)

 

f(x)  =  f(a) + f’(a) (x-a)/1!  +  f’’(a) (x-a)2/2!  +  . . . f n(a) (x-a)n/n!    

 

where  f n(a)  is the nth derivative of f(x) evaluated at x = a

 

 

Another way to get a series expansion of a function (i.e. inverse functions)  is to

integrate the derivative of the inverse function.

 

               t = x

    f(x)  =  ∫    [df/dt]  dt                                        Click here for an example.

              t = 0

 

 

Rule of Thumb

 

To estimate the number of terms in a series for a specified accuracy, use two more decimal

places in the computations than required for the final answer.

 



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