STEP III 2007 question 2 solution

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(i)

as required.

Binomial expansion valid for i.e. as given.

Every successive term is multiplied by two extra negative terms, one fraction from the exponent and one power of -4x, so every term stays positive overall, so all minus signs can be removed.

Using the formula just proved and noting that there is a divisor of in total from the product of :

as required.

Verify that 2x is the first term by setting n = 1 and getting 2x.

(ii)

Differentiate both sides with respect to x:

Setting :

Multiply both sides by :

as required.

(iii)

This time integrate instead:

Set x = 0:

Set x = :

as required.

Solution by Speleo.