First part :


(x² + 2x) / (2x² + 1)

= (x² + 2x) (1 + 2x²)-¹

= (x² + 2x) (1 + (-1)(2x²) + (-1)(-1 - 1)/2! (2x²)² + ...

(Based on MF26 formula for (1 + x)ⁿ . In this case we sub in 2x² as our x. We want up to power 5 only so we can stop at the 3rd term)


= (x² + 2x) (1 - 2x² + 4x⁴ + ...)

= x² - 2x⁴ + 4x^6 + 2x - 4x³ + 8x^5 + ...

≈ 2x + x² - 4x³ - 2x⁴ + 8x^5



For the expansion to be valid,

| 2x² | < 1
2 | x² | < 1
| x² | < ½
| x |² < ½
| x | < √½

-1√2 < x < 1/√2
√2 / 2 < x < √2 / 2

Second part :

The terms in the expansion of (1 + 2x²)-¹ all have even powers of x. The power of x is increasing by 2 each time.

To get x^2017, we should look for the term in x^2016 first.

The product of this term and the 2x in the left bracket will give us a term in x^2017.

Since the term is 2x² inside the bracket, we should look for the 1008th power in order to get a term in x^2016 .


Term in x^2016

= (-1)(-2)(-3)...(-1007)(-1008)/1008! (2x²)^1008

= (1)(2)(3)...(1007)(1008)/1008! (2^1008) x^2016

(All the negatives cancel out since there is an even number of factors with a negative sign)

= 1008!/1008! (2^1008)x^2016
= 2^1008 x^2016


What we also realise is that there is actually no need to write out all the multiplication since we observe a pattern in the expansion:

①The sign in front alternates. Odd powers of (2x²) have - , even powers have +
② The coefficient is just 2ⁿ, where n is the power that (2x²) is raised to.

So for term in x^2016, we could immediately write 2^1008 x^2016 since the power is even for (2x²)^1008


Term in x^2017

= 2x (2^1008 x^1006)
= 2^1009 x^1007

Coefficient of x^2017

= 2^1009

Alternatively,

We could already see a pattern in 2x + x² - 4x³ - 2x⁴ + 8x⁵ for the odd powers of x :

Rewrite the odd terms as :

(-1)^(½(1-1)) 2^(½(1+1)) x¹
(-1)^(½(3-1)) 2^(½(3+1)) x³
(-1)^(½(5-1)) 2^(½(5+1)) x⁵


We observe the odd powers of x and derive the general formula for a term in xⁿ where is odd :

Term in xⁿ = (-1)^(½(n-1)) 2^(½(n+1)) xⁿ

So for term in x^2017, the coefficient would be :

(-1)^(½(2017-1)) 2^(½(2017+1))

= (-1)^(½(2016)) 2^(½(2018))

= (-1)^1008 2^1009

= 2^1009

Thank you very much for your help.

Welcome

Edit : for the inequality, it should be -√2 / 2 on the left side instead of √2 / 2