Description
- In lectures we proved the result
via a somewhat circuitous route. Prove this result instead by using the principle of mathematical induction.
- 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 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.)
- Find the Taylor series for f(x) = ln(x), centred at a = 3, along with its interval of convergence.
- Consider the series, with terms an 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.
- Use Maclaurin series to compute the limit
.
