Why Is the Binomial Expansion Only Valid for |a| < 1?

[phospho]

http://www.examsolutions.net/maths-revision/core-maths/sequences-series/binomial/formula/validity/tutorial-1.php

On the above video, he states that the binomial expansion is only valid for |a| < 1 when n is not a positive integer. I understand that when n is not a positive integer the expansion will be infinite as no coefficient will ever be 0, however I don't understand why |a|< 1 ? What if a was 2, the expansion would still be valid no?

 

[dextercioby]

If a is not between -1 and 1, the series will not converge. Remember, n is fixed. The convergence will be dependent on a.

 

[phospho]

If a is not between -1 and 1, the series will not converge. Remember, n is fixed. The convergence will be dependent on a.

How is it dependent on a? Is it similar to infinite geometric series where |r| < 1, where r is the common ratio?

 

[Ray Vickson]

http://www.examsolutions.net/maths-revision/core-maths/sequences-series/binomial/formula/validity/tutorial-1.php

On the above video, he states that the binomial expansion is only valid for |a| < 1 when n is not a positive integer. I understand that when n is not a positive integer the expansion will be infinite as no coefficient will ever be 0, however I don't understand why |a|< 1 ? What if a was 2, the expansion would still be valid no?

No, it would not. If a > 1 the expansion is not valid; if a < -1 the expansion is not valid. Try it for yourself: take n = 1/2 and write out a few of the terms for a = 2 and for a = -2. Note that for a = -2 we have (1+a)^(1/2) = sqrt(-1) = i, the pure imaginary, but all the terms in the binomial expansion are real.

RGV