Applications using the Binomial Theorem

 

 

Example:    Find the power series expansion of  f(x) =  4 / ( 5 + x )3   

                     and determine its radius of convergence.

 

 

 

Recall the binomial series expansion:

 

( 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     

 

 

Rewrite:      f(x) =  4 / ( 5 + x )3  =  (4/53) 1 / (1 + x/5)3  =  (4/53) (1 + x/5) ˗3    

 

By the binomial theorem: 

 

        (1 + x/5) ˗3  =  1  ˗ 3[ (x/5) + (˗4)(x/5)2 /2!+ (˗4)(˗5)(x/5)3 /3! +  .  .  .  ]

 

                                           ∞

So               f(x)  =   (4/53) ∑ ( { ˗ 3 / n }  (x/5)n        ( result for series of   f(x) )

                                          n=0

where   { ˗ 3 / n }  =  [ (˗ 3 ) (˗ 4 ) (˗ 5 )  .  .  .   (  ˗ 3  ˗  n  + 1 ) ] / n!

 

Next find the radius of convergence of the power series, f(x).

 

 Strategy:  Apply       lim    | f(x) n+1 / f(x) n |  

                                 n → ∞

 

=   lim | ([(1)(2)(˗3)  . . (˗3˗n+2) ] (x/5)n+1/ 2( n+1)!] / [ [1)(2)(3)  . . (˗3˗n+1) (x/5)n / 2 n! ]  |

  n → ∞

 

=   lim | ([ (˗3 ˗ n + 2 ) (x/5) / ( n+1 )]  | <  1   for convergence

  n → ∞

 

=   lim |  (x/5)  |  <  1   for convergence   or   lim | x | <  5      So   R = 5     (result)

  n → ∞                                                         n → ∞

 

 

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