Find more, included in the manuscript on integration by parts.
Some (not so popular, but feasible) approaches for teaching calculus would
cover the ex function and its properties only after you have learnt
about power series. They would come up with a new function E(x), defined
by the formula
E(x):= 1+x+x2/2! + ... + xn/n! + ...
and you would not have any information on the properties of this function yet. The (somewhat sophisticated) task would then consist of retrieving all properties from this power series. Here is one instance of this type of task: By multiplying the series with itself, show that indeed E(x)2 = E(2x). If you find it too difficult to evaluate the general (nth) term of the series for E(x)2, content yourself with showing that the first 5 terms of the series on each side of the equation coincide.
Write down the McLaurin series (=Taylor series at 0) for sinh x and compare it with the one for sin x. (Indeed, some problems can be both easy and nutritious :-)
Calculate limx->0 (tan x - sin x) / x (1-cos x)
(a) using l'Hospital
(b) using Taylor series
Compare the amount of work for either approach.
Find first the number k such that
limx->0 (3 sinh x + (2 cos x - 5) sin x) / xk
is neither 0 nor infinity, then find the limit with this choice of k.
This is actually a rather natural type of question. In any kind of problem, you may come up with the answer given as a formula whose consequences you do not understand immediately. Let's assume f(x) = 3 sinh x + (2 cos x - 5) sin x is such an answer. How does the function graph look like? Here, we are concerned with the question how f(x) behaves for small x. Once you see that f(0)=0, you conclude that f(x) will be small, if x is small. But how small, in relation to x, will f(x) be? More like x or more like x2 or what else? The above problem will answer exactly this type of question.
Sumn=0infinity xn/2n and make sure for which x the evaluation holds.
(b) Using the result, evaluate now Sumn=1infinity n xn-1/2n
(c) Continuing, what is the numerical value of Sumn=1infinity n /2n ?
(d) Now cover up the first to parts of this problem and imagine you had been given the third part alone. You would have needed to invent the first to parts yourself, as auxiliary steps.
(e) Having savored the difficulty and solution of the dilemma described in (d), what is the numerical value of Sumn=1infinity n2/2n ?
Explain the difference between conditional and absolute convergence of a
Which of the following sequences is [ nonincreasing | nondecreasing | bounded
(-1)n - cos (pi) n , where (pi) stands for the greek letter pi = 3.14... (pardon my ignorance how to typeset this in html)
Does each of the following series converge (absolutely or conditionally?) or
diverge? Explain why (which test) and be sure to check all conditions of the
Sumn=1infinity (-1)n (n+5)/2n
Sumn=1infinity ln n / n
Sumn=0infinity (n^2 + 77) / (n! - 33)
Does the improper integral converge or diverge? Integral1infinity dx / (x2+sin x)