MS-A0111 – Differential and Integral Calculus 1 Problem set 1

2025–2026 (Period I)
Milo Orlich – Johan Dinesen Deadline: Sun 7.9.2025 at 23:59
Submit on MyCourses your solutions to Homework 1 and 2 by Sunday, September 7.

Warm-ups
Work on Warm-up 1–4 during the exercise sessions of Week 1. You do not need to submit solutions
to these problems.
Warm-up 1. Show explicitly (going through the definition of limit with ∀ and ∃) that:
en + 2
(a) lim =1 (b) lim (n2 − n) = +∞.
n→+∞ en n→+∞

Compute the following limits, if they exist:

5 − 2n n3 − 4 en − e−n
(c) lim (d) lim (−1)n (e) lim .
n→+∞ 3n − 7 n→+∞ n2 + 5 n→+∞ en + e−n

Warm-up 2. For each point, give an example or explain why an example does not exist:
1. Sequences (an )n∈N and (bn )n∈N that do not converge and such that the product sequence
(an bn )n∈N converges.

2. A sequence (an )n∈N that does not converge and such that the sequence (|an |)n∈N of ab-
solute values converges.

3. Sequences (an )n∈N and (bn )n∈N such that

lim an = +∞, lim bn = 0 and lim (an bn ) = 5.

n→+∞ n→+∞ n→+∞

4. Sequences (an )n∈N and (bn )n∈N such that

lim an = +∞, lim bn = +∞ and lim (an bn ) = 5.

n→+∞ n→+∞ n→+∞

Warm-up 3. Find the value of the following series:

+∞ +∞
X 1 X 1
(a) (b) .
2 · 2n n(n + 1)
n=2 n=1

Hint for (a): Rewrite it to get a geometric series. Hint for (b): Compare the terms an and an+1 . Write
explicitly the sum SN = a1 + a2 + · · · + aN , and use the definition of a series.
Determine whether the following series converge.
+∞ n+1 +∞ 5 +∞ +∞
X 2 X n X n! X n2
(c) (d) (e) (f ) .
nn 2n nn n2 − 3
n=1 n=1 n=1 n=1

Warm-up 4. For each of the following power series, determine all the values of x for which
the series converges. In each case, if the convergence does not hold for all x ∈ R, what are
the center and the radius of convergence?
+∞ +∞ +∞
X (1 + 5n ) n
X
3 n
X 1
(a) x (b) n (2x − 3) (c) (x + 2)n .
n! 2n n
n=0 n=0 n=1

1
Homework
Submit on MyCourses your solutions only for the following two problems. Explain the reasoning
behind your solutions, do not just return the final result. If you use important results from the
lectures (for instance a theorem or a famous limit), state what they are.
Homework 1. Compute the following limits, including all the steps in your calculation.
n √
(a) lim (c) lim n − n2 − n
n→+∞ ln(n2 + 1) n→+∞

n−1 n n5 + en + 4n4
 
(b) lim (d) lim 3 n .
n→+∞ n + 1 n→+∞ n − e + 2n2

A+B
Hint: If A + B , 0, you can write A − B = (A − B) A+B . [2 points]
P+∞ n P +∞ n
Homework 2. Given two power P+∞ series n=0 cn x and n=0 dn x , we define their product to
be the power power series n=0 an xn , where the n-th term is
X
an = c0 dn + c1 dn−1 + c2 dn−2 + · · · + cn−1 d1 + cn d0 = ci dj .
i+j=n
i,j≥0
P+∞
(a) P
Write the product of the power series 1+x+x2 +· · · = n=0 x
n and 1−x+x2 −x3 +x4 +· · · =
+∞ n n P +∞ n
n=0 (−1) x in the form n=0 an x , for suitable an .

(b) What is the interval of convergence of the product in part (a)? To what function of x is
the product equal to?
[2 points]

Additional problems for extra practice

The following problems will NOT be discussed during the exercise sessions and are NOT for homework.

Extra problem 1. Draw by hand, without the help of a computer, the following functions:

f (x) = |2x − 1| − 1, g(x) = e|x|−1 , h(x) = x2 sin(πx).

If the graph of the function f (x) is as in the following drawing, what do the graphs of the functions f (x) + 2
and f (x + 2) look like?

Extra problem 2. Translate the following formulas into English statements and determine whether they are
true or false:

(a) ∀x∈R x < −1 (d) ∀x∈R ∀y∈R x + y = xy (g) ∀x∈R ∀y∈R |x + y| = |x| + |y|
(b) ∃x∈R x < −1 (e) ∃y∈R ∀x∈R x + y = xy (h) ∀x∈R ∀y∈R |xy| = |x||y|
(c) ∃x∈N x < −1 (f ) ∀x∈R\{1} ∃y∈R x + y = xy (i) ∃x∈R |x| = x + 1.

Extra problem 3. Compute the following limits:

e2x − 1 sin(ln(1 + x2 )) 3 x x5 + 4x4 + ex

 
(a) lim (b) lim (c) lim 1 + (d) lim .
x→0 ln(x + 1) x→0 x x→+∞ 2x x→+∞ x3 + 2x2 − ex