[Solved] MATHS7027 Assignment 2

1. In lectures we proved the result

via a somewhat circuitous route. Prove this result instead by using the principle of mathematical induction.

2. In lectures we derived an expression for the Maclaurin polynomial for cos(x).
   • Using this expression, find the Maclaurin polynomial of degree n = 2k for f(x) = cos(2x).
   • Use Taylors theorem to estimate how many terms need to be used to approximate cos(2) to within 0.001. (Hint: For f⁽ⁿ⁺¹(z), think about what y-values cos(x) and sin(x) are both bounded by. Youll need to use some trial and error to find n once you have a bound for the remainder.)
3. Find the Taylor series for f(x) = ln(x), centred at a = 3, along with its interval of convergence.
4. Consider the series, with terms a_n defined recursively by the equations:

for some given value of k N.

• Write out the first 6 terms of the series (i.e., up to n = 5).
• Use the ratio test to show that the series converges for k 5, and diverges for k 3.

5. Use Maclaurin series to compute the limit

.