How do you use the binomial series to expand # 1/((2+x)^3)#?

Answer 1

The binomial expansion is
#= 1/8 (1- 3/2 x+3/2 x^2-5/4 x^3+...)# for #|x|<2# and

#= 1/x^3 (1-6/x+24/x^2- 80/x^3+...)# for #|x|>2#

One important point to remember is that if the index #n# is not a positive integer, the binomial expansion leads to an infinite series. In such a situation, it is important to keep in mind that the infinite binomial series for #(1+x)^n# converges only if #|x|<1#.

For an arbitrary #n#, the binomial expansion for #(1+x)^n# is given by

#1+n x+ {n(n-1)}/{2!} x^2+ {n(n-1)(n-2)}/{3!} x^3 +... #

So we first recast the expression that we want to expand binomially in the form #(1+x)^n#

Here we first use #1/{(2+x)^3} = 1/2^3 (1+x/2)^{-3} #

and then use the binomial expansion for #n=-3# to get

#1/{(2+x)^3} = 1/2^3 (1+(-3) x/2 + {(-3)(-4)}/{2!} (x/2)^2 + {(-3)(-4)(-5)}/{3!} (x/2)^3 +.... )# #= 1/8 (1- 3/2 x+3/2 x^2-5/4 x^3+...)#

Of course this infinite series only converges for #|x/2|<1#, i. e. #-2 < x <2#.

If #|x/2|>1#, this series will diverge - rendering it useless! What saves the day is that in this case, #|2/x|<1# and so, we can still get a binomial expansion, but thus time we must start from #1/{(2+x)^3} = 1/x^3 (1+2/x)^{-3} # and continue with

#1/{(2+x)^3} = 1/x^3 (1+(-3) 2/x + {(-3)(-4)}/{2!} (2/x)^2 + {(-3)(-4)(-5)}/{3!} (2/x)^3 +.... )# #= 1/x^3 (1-6/x+24/x^2-80/x^3+...)#

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email

Answer 2

Caleb Alexander

To expand ( \frac{1}{(2+x)^3} ) using the binomial series, you can follow these steps:

Identify ( a ) and ( n ) in the general form of the binomial series: ( (1 + x)^n ).
- ( a ) is the constant term, which is 2 in this case because ( (2 + x)^3 ) can be rewritten as ( (2(1 + \frac{x}{2}))^3 ).
- ( n ) is the power to which the binomial is raised, which is 3.
Apply the binomial series formula: [ (1 + x)^n = \sum_{k=0}^{\infty} \binom{n}{k} a^{n-k} x^k ]
Substitute ( a ), ( n ), and ( x ) into the formula.
Expand the terms using the binomial coefficients ( \binom{n}{k} ), which are also known as combinations.
Simplify the expression to get the expanded form of ( \frac{1}{(2+x)^3} ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.