You may want to add a part (c) to this: Draw a sketch (function graph in coordinate system, plus whatever is appropriate in this graph) such that the result in 47b becomes geometrically obvious, even without calculation.

Find more, included in the manuscript on integration by parts.

Some (not so popular, but feasible) approaches for teaching calculus would
cover the e^{x} 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+x^{2}/2! + ... + x^{n}/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
(n^{th}) 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 lim_{x->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
lim_{x->0} (3 sinh x + (2 cos x - 5) sin x) / x^{k}
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 x^{2} or
what else? The above problem will answer exactly this type of question.

(a) Evaluate
Sum_{n=0}^{infinity} x^{n}/2^{n} and make sure for which x the evaluation holds.

(b) Using the result, evaluate now
Sum_{n=1}^{infinity} n x^{n-1}/2^{n}

(c) Continuing, what is the numerical value of
Sum_{n=1}^{infinity} n /2^{n} ?

(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
Sum_{n=1}^{infinity} n^{2}/2^{n} ?

Explain the difference between conditional and absolute convergence of a

Which of the following sequences is [ nonincreasing | nondecreasing | bounded
convergent ]?

(-1)^{n} - cos (pi) n , where (pi) stands for the greek letter
pi = 3.14... (pardon my ignorance how to typeset this in html)

n^{n}/n!

(n+2)/(n+1)

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
relevant test(s).

Sum_{n=1}^{infinity} (-1)^{n} (n+5)/2^{n}

Sum_{n=1}^{infinity} ln n / n

Sum_{n=0}^{infinity} (n^2 + 77) / (n! - 33)

Does the improper integral converge or diverge?
Integral_{1}^{infinity} dx / (x^{2}+sin x)