MATH3503 Online Assignment 12

Make sure to use the link on the Learning Hub, which contains a user name and a user ID unique to you, otherwise your mark would not be recorded. You may use the URL without '?uname=...&uid=...' if you just want to practice. You will get different numbers every time you open the page.
Numerical answers are evaluated as correct if the difference from the correct value is less than `0.001 times` (correct value) unless otherwise specified. That means, you need to keep at least 5 significant digits (e.g., 12.34, 0.01234, etc.) in your final answer, and a couple of more digits in the intermediate steps.

Infinite series

Complete the following convergence tests.
1. \(\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{1.2}}\) converges because \(\displaystyle\int_1^\infty \frac{dx}{x^{1.2}} =~ \)
2. \(\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{0.9}}\) diverges because \(\displaystyle\lim_{k\to\infty}\int_1^k \frac{dx}{x^{0.9}} = \lim_{k\to\infty}\)\(=\infty\)
Consider a geometric series \[ \sum_{k=1}^{\infty}(4x + 1)^k. \]
1. The range of convergence is \( \lt x \lt \)
2. If `x=-0.3` then, \(\displaystyle\sum_{k=1}^{\infty}(4x + 1)^k = \)
Find the Taylor expansion of the following functions about the specified value of `x=x_0` to third order in `x` (i,e, truncate terms after the one containing `(x-x_0)^3`).
1. About `x=0` \(\displaystyle\frac{1}{1-7x} \approx~ \)
2. About `x=0` \(\displaystyle\ln(4+x) \approx~ \)
3. About `x=1/5` \(\displaystyle\ln(5x) \approx~ \)
4. About `x=0` \(\displaystyle e^{-4x^2} \approx~ \)
5. About `x=0` (Enter pi for `pi`.) \(\displaystyle \sin\left(\frac{\pi x}{5}\right) \approx~ \)
6. About `x=-2.5` \(\displaystyle \cos\left(\frac{\pi x}{5}\right) \approx~ \)
The restoring force (i.e., `f(0) = 0`) of a retaining wall is modeled as \(f(x) = 7x e^{-2x}\).
1. Find the linear approximation of `f(x)` about `x=0`. \(\displaystyle f(x) \approx~ \)
2. To compute the error bound (\(|R_1|\le\frac{M x^2}{2!},~ M=\max_z f''(z)\)) for this linear approximation, first compute \(f''(x)\). \(f''(x) = ~\) *Enter a function of `x`.
3. The maximum of \(f''\) occurs at `x` that satisfies \(f'''(x) = 0\) (You did it in Calc. 1!) \(\max f''(x) = ~\) *Enter a number.
4. Thus, the remainder is bounded; \(|R_1| \le ~\) *Enter a function of `x`.
5. Finally, the range of `x` where the linear approximation is valid can be computed from the fact that the linear term (obtained in a.) is much larger than the error bound for `R_1` found in d. \(|x| \ll ~\)