Binomial series for rational powers
The Pure 1 binomial expansion becomes an infinite series when the power is negative or non-integer. You need only the requested first few powers.
Build the first terms from falling factors
Build one coefficient at a time
x⁰ term
1
x¹ term
nx
x² term
n(n−1)x² / 2
x³ term
n(n−1)(n−2)x³ / 6
for non-integer or negative n: |x| < 1
- Substitute the numerical value of immediately.
- Stop at the highest power requested.
- A general term is not required by the 9709 syllabus.
Factor to the form one plus something
Worked example
Current-paper square-root expansion
Use and replace the series variable by .
Answer
(9709/32/O/N/24 Q1)
Common mistake
The outside factor 3 multiplies every term, including the term.
Negative powers do not terminate
For , the falling factors are ; none becomes zero. The dots are part of the answer because the series continues.
Your turnIndependent check [4 marks]
Expand up to and including the term in .
Show worked answer
Substitute .
Answer