Binomial Expansion Part2

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Contents 1 Binomial Coefficient Revisited 2 Expansion for Non Positive integer n 2.1 Important Differences 2.2 Of Apples & Oranges & Bananas 2.3 Calculating Coefficients 2.4 Examples 3 Things To Do Before Expanding 3.1 Determining value of n 3.2 Changing Into (1+x)^n 3.3 Multiple Expansions 3.4 Example 4 Comparison 5 Validity 5.1 Simplifying the Validity Range 5.2 Identifying the x 5.2.1 Examples 5.3 Stating the Validity 6 Exercise 5 7 Ascending powers of 1/x 7.1 Example 8 Finding value of n 9 Exercise 6

Binomial Coefficient Revisited

Let's look at the first few binomial coefficients

• 
• 
• 
• 
• Can you guess the value from the pattern of the first few?
• And the next one will be

Expansion for Non Positive integer n

• Valid for

At a glance, it seems to be the same formula we have already learnt for positive integer , just with the binomial coefficients written in the other form (refer above), but there are fundamental differences between the two.

Important Differences

• Non positive integer here includes
  • negative integers, fractions, negative fractions
• It is an infinite series (never ending), and we MUST write at the end. Normally we would be told which term we need to expand until.
  • Compare with positive integers
    • will end at
    • will end at (but we might be asked to expand until certain terms only)
    • will have a never-ending expansion (but we WILL be told to expand till where)

• We MUST use form, there is no form for non positive integer
  • For positive integers we can use either.
• Expansion is only valid for
  • For positive integers it is valid for all values.
• For non positive integers, does not exist (your calculator will give an error), thus we MUST use the other representation of the coefficients.
  • However, can we use that alternative representation for positive integer? Yes.

Of Apples & Oranges & Bananas

Actually, there are only two variables here, (the is fixed). Lets see how we are going to use the formula.

• 

• 

• So we always start with
• Followed by "", which is the power times whatever we use as ""
• The following terms,
  • the "" we just keep increasing the power
  • the "", follows a set pattern, each term having "more" compared to the previous

Calculating Coefficients

Lets say we have ,

• the coefficient will become , which, though correct, is messy to calculate, not to mention creates a high risk of careless mistake
• a better way to write it is to first understand as series of steps
  • take
  • minus one and multiply it
  • minus one again and multiply it
• Thus, we will write
  • then minus which will give us
  • then take and minus again which will give us
  • 
  • Thus
  • In fact, note that we would have already written earlier, thus we actually do one extra step only

Note : Again, the pattern of the signs is either

1. all positive
2. all negative (except first term)
3. positive and negative alternately

Examples

Expand the following in ascending powers of until the term in

• 
• • Let's write it out carefully
    • 
    • DON'T write yet.
    • 
    • Notice how we just keep on "expanding" the coefficients from previous one.
    • Finished? Sure? Must put
  • 
  • 

• 
• • calculate the fractions carefully
    • 
    • 
    • 
    • 
  • To calculate using calculator, rather than typing the whole thing (waste time and easy to press wrongly), we might as well take (product of numerators) divide by (product of denominators) and then divide by , and then determine the sign.
  • Note: Write instead of

Try these on your own :

• Answer :
  • If you got it wrong, it's probably because while paying attention to the your forgot about the "apple".
  • 

• Answer :
  • 

Things To Do Before Expanding

Usually, there are a few steps that need to be done before we can even start using that formula. Do note that if there are mistakes in these early steps, everything else will be wrong and just a waste of time.

Determining value of n

Make sure you don't make a mistake in this step

• 
• Note: There is no reason to disturb the , just leave it as it is.
• Note: There is, again, no reason to disturb the
• 
• Note: and NOT
• 

Changing Into (1+x)^n

In the formula

• it MUST be a
• "" can be anything

So what do we do if the question is NOT a ?

Suppose we have , we can change the to a by factorizing the .

• 

What if we have  ?

• 
  • Now, the will look troublesome, but remember we DON'T have a choice, we MUST make the first term to be

But the above is without any power. How will affect it?

Remember that

In other words, when we want to bring the out, we must apply the power to it too.

• 

Try these

• 
• 
• Note:
• 

Multiple Expansions

Keep these in mind

• its always easier to than
  • Thus, if applicable, we would expand the partial fraction form
• it is very difficult to divide polynomial
  • Multiplying, while a lot of work, is still doable. But how do we change divide into times?

• 
• 
• 
  • We could have expanded top and bottom, but we will end up with a polynomial divide polynomial.
• 
• 

Example

Let's put all these to work:

Expand the following in ascending powers of , neglecting the terms in and above

• 
• • Analyze : There is no point expanding top and bottom. Instead, we must first rearrange it.
  • 
  • Since both expansion is going to require some work, better we do it separately first
  • Analyze : We need to do some work first since it is not
    • Now we can expand. What about the ? Just leave it outside first.
    • 
    • 
    • 
  • Analyze : It's already , so we just expand
    • 
    • 
  • Next step would be multiplying them. Note: DO spend some time checking both expansions before multiplying.
  • 
  • 
  • 
  • Valid for Read the part on validity (below) first and then come back and try
    • and
    • and
    • 
    • 

Comparison

As always, let's take a breather to compare the different things we have learnt thus far.

• For positive integer
  • 
  • 
    • 
• For non positive integer
  • 
  • Valid for
• If the first term is NOT
  • For positive integer  : Don't need (but can) change it to
  • For non positive integer  : MUST change it to
• Coefficients
  • For positive integer  : Can use or form
  • For non positive integer  : MUST use form
• Other difference
  • For positive integer  : Finite expansion. Valid for all values
  • For non positive integer  : Infinite expansion. Valid for

Validity

Let's have a look at the formula again

• 

The expansion is an infinite series, thus as , we must have (so that the series converges)

Note : Otherwise, we will keep on adding more and more terms which is increasingly large, thus (which is of course wrong). We can compare this with geometric series, where sum to infinity only exists if , so that as ,

For as to be true, we must have

Thus, the expansion is only true when

We say the expansion is valid only for or .

Important

• The "" here refer to WHATEVER we use ""
• Final answers are normally written in the form, but the modulus form is useful to simplify

Simplifying the Validity Range

• Simplify the modulus form FIRST before changing to the other from
  • 
  • (modulus takes care of the negative sign already)
• Since is always positive, we CAN multiply it across
  • or
  • or
• is always positive ( actually)
  • (modulus sign is not needed anymore)
  • We can't take away the modulus for this one, however, and this would need to be solved slowly using normal inequalities method.
• Combination
  • and
  • and
  • and and

Identifying the x

The "" that we are going to put into is the same whatever "" we use in the expansion. In other words

• 
• Valid for

Examples

• 
  • Valid for
• 
  • Valid for
• 
  • Valid for
• We must see from this step, right before we use the formula (rather than the earliest form)
  • Valid for
• 
  • Valid for and
• 
  • Valid for and
• The first bracket DON'T need since we did not expand it
  • Valid for
• The first bracket DON'T need since it's expansion for positive integer (which is valid for all)
  • Valid for
• 
  • Valid for

Stating the Validity

What form we state it depends on the form of the question

• Question : State the range of where/such that the expansion is valid
  • Answer : Valid for
• Question :State the set of values of where/such that the expansion is valid
  • Answer : Valid for (without set notation, marks won't be given)
• Question :Find set where the expansion is valid when
  • Answer : Valid for (just a fancy way to ask you for the set)

Normally, the question will state it clearly if you are required to state the validity range. If not asked to do so, you need not state it, but you might want to write it out for your own reference when doing approximation later.

Exercise 5

Notes

• Certainly a lot of things here to be taken care of. Make sure you are very familiar with all the above before attempting this.

1) Expand the following in ascending powers of until the terms in and above.

• a)
• 

• b)
• 

2) Expand the following in ascending powers of , neglecting the terms in and above. State the set of values of where the expansion is valid.

• a)
• • 
  • 

• b)
• • 
  • 

• c)
• • 
  • 

• d)
• • 
  • 

• e)
• • 
  • 

• f)
• • 
  • 

• g)
• • 
  • 

• h)
• • 
  • 
  • 
  • Media:MT-BE-E5-2h.dia
  • 
  • 

Ascending powers of 1/x

Note that is in ascending powers of

Whereas something like will be in ascending powers of

So how would we get an expansion which is in ascending powers of ? (Question will state clearly if it is required)

• 

Note:

• Thus we would need to change into the above from before expanding
• Ascending powers of can sometimes stated as descending powers of , since means

For example, if we are required to expand in ascending powers of

• 
  • Normally (ascending powers of ), we will just rearrange it to , but that won't work here.
  • Instead, we need to somehow have
  • In this case, we force it by factoring an out.
  • 
  • 

Valid for :

• 
  • 
  • 
  • 

Valid for :

Example

Express in partial fractions. Hence, find it's expansion in ascending powers of till the term in . State the range where the expansion is valid.

• • First, using partial fractions (we will not show the working here), we will get
    • Note : It is IMPORTANT to check that the partial fraction is correct before continuing
  • Noting that we are required to expand in ascending powers of , let's expand the two terms separately first
    • 
    • 
    • Not that since there is a outside the bracket, we need to expand to only
    • 
    • 
    • 
    • 
    • 
    • 
    • 
    • 
    • 
  • 
    • 
  • 
    • 
    • 
    • 

Finding value of n

When we are given the expansion (or some other information) and asked to find the value of ,

• Just expand like normal, coefficients will be in terms of
• Compare with the given information and solve
• Check that your final value satisfies the information

Example 1: If the first three terms of the expansion of is , find the values of and .

• 
  • 
  • 
  • 
  • We just compare the coefficient, DO NOT write
  • 
  • 
  • 
  • Check :

Example 2: In the expansion of , given the coefficient of the term in is two times the coefficient of the term in , find value of which is a positive integer.

• Analyze : Since is positive integer, we CAN use the formula first, rather than having to make the first term to be
• 
  • Alternatively, we can also use the formula
  • But & are difficult to manipulate, since we don't know the value of , thus we change it to the alternative form, either by simplifying the factorials or simply refer back to the formula for non positive integer
  • 
  • Let's use the information given, keep in mind we only equate the coefficients, not the term itself (don't write the )
  • Let's deal with all the numbers first.
  • But
  • 
  • 
  • 
  • 
  • 
  • 
  • Check :
  • Alternatively,
    • 
    • 
    • 
    • 
    • 

Exercise 6

Notes

• These are pretty hard question, but there are similar to actual STPM questions. You can try to redo the examples again on your own first to get the hang of it before attempting.

1) Express in partial fractions. Hence, or otherwise, find the first three terms in the expansion

• a) in ascending powers of
• b) in ascending powers of

For each expansion, find the range of values of such that the expansion is valid.

• • 
  • 
  • 
• a)
  • 
  • 
• b)
  • 
  • 

2) If the first three terms of the expansion of is find the values of and .

• • 

3) Find the value of which is a positive integer, if

• a) In the expansion given the coefficient of the term in is four times the coefficient of the term in , which is a positive integer.
• • 

• b) In the expansion given the coefficient of the term in is five times the coefficient of the term in .
• •