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Contents
Binomial Series
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Binomial Theorem
for positive integral index
The binomial series is an extension
of the binomial theorem, such that it can be used for negative or non-integral
powers. However, one condition must be satisfied.
Condition: |x|<1
The binomial series is derived from
the Maclaurin's Theorem. It is an infinite series. Note that the first term of
the binomial must be 1.
 Expand
in ascending powers of x as for as the term in x3 and state the range
of x for which the expansion is valid.
(i)
(1 - 2x)-2
(ii) (1
+ x)1/2
(iii) (2 + x)-2
Solution:
(a) (1 - 2x)-2
Expansion valid when |2x| < 1
---> -0.5 < x < 0.5
(b) (1 + x)1/2
Expansion valid when |x| < 1
---> -1 < x < 1
(c) (2 + x)-2
Expansion valid when |x/2| < 1
---> -2 < x < 2
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