### Chapter 7.1; Algebraic Integration Techniques

Any nutritious hwk I have come up with here has been integrated in the manuscript art of integration.

### Chapter 7.2; Integration by parts

See #47 in the textbook (Feb 23)
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.

### Chapter 7.x; Any integrals

As it looks to me, numbers 1-6, and 12,13 from last year's exam are appropriate for practising the material you expect to see here. For practice purposes you may request your roommate to copy them for you *without* the hints given in the exam.

### Chapter 8.6-8; Power Series

Using the Taylor series for ex at x=0 and only the plus, minus, times, and divide keys on your pocket calculator, calculate e-20 with a precision of 7 decimals. --- Hint: e-20 = (e-0.625)32

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.

(a) Evaluate 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 ?

### Chapter 7.7-8.6; Some training problems for the exam

Does the sequence {an} given by an= ((n3+1)/(n4+1))1/n converge? Explain why it does or does not. If it converges, calculate its limit.

Explain the difference between conditional and absolute convergence of a sequence (This is a trick question) Answer

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)
nn/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).
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)