[A more general form of the binomial theorem] a Show that the binomial expansion can be written as n (n – 1)(n – 2), п (п — 1),2 (1 + x)" = 1 + nx + 2! 3! b In this form, it can be shown that the expansion is true for negative or fractional values of n, provided that the RHS is regarded as the limit of an infinite sum of powers of x. This is called the power series expansion of (1 + x)". Assuming this, generate the binomial expansions of: iv v1 + x i ii (1 – x)? (1 + x)? c Verify, using your expansions in parts bi and ii, that dx \1 - x) (1 – x)?" (This assumes that a power series can be differentiated term-by-term.)
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b and c
![[A more general form of the binomial theorem]
a Show that the binomial expansion can be written as
n (n – 1)(n – 2),
п (п — 1),2
(1 + x)" = 1 + nx +
2!
3!
b In this form, it can be shown that the expansion is true for negative or fractional values of n, provided
that the RHS is regarded as the limit of an infinite sum of powers of x. This is called the power series
expansion of (1 + x)". Assuming this, generate the binomial expansions of:
iv v1 + x
i
ii
(1 – x)?
(1 + x)?
c Verify, using your expansions in parts bi and ii, that
dx \1 - x)
(1 – x)?"
(This assumes that a power series can be differentiated term-by-term.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F211fd0d2-45cd-4603-b99f-222084379316%2F85ddd035-8f98-40c5-a62e-63d752045529%2Ffeq73l6.png&w=3840&q=75)
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