MATHS 7027 Mathematical Foundations of
Data Science
Assignment 2
(PDF only)
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).
(a) Using this expression, find the Maclaurin polynomial of degree n = 2k for f(x) = cos(2x).
(b) Use Taylor’s 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. You’ll 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 an defined recursively by the equations:
for some given value of k ∈N.
(a) Write out the first 6 terms of the series (i.e., up to n = 5).
(b) 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
.
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