1 Mathematics II

Occasionally the lim will be omitted throughout the script, but the fact that it is a limit will be clear by the comments.

1.1 General Assumptions

• For equations which contain infinity or zero, the conclusion wether the result is zero, infinity or undefined

is often hard to make: 1. ∞ ± a = ∞ ∀a ∈ ℝ

2. ∞ + ∞ = ∞

3. ∞ − ∞ is undefined

4. a · ∞ = \{ \begin{matrix}∞ & for & a > 0 \\ −∞ & for & a < 0 \\ undefined & for & a = 0 \end{matrix}

5. \frac{∞}{a} = \{ \begin{matrix}∞ & for & a > 0 \\ −∞ & for & a < 0 \end{matrix}

6. \frac{a}{∞} = 0 ∀a ∈ ℝ

7. ∞ · ∞ = ∞

8. \frac{∞}{∞} is undefined!

9. \frac{a}{0} = \{ \begin{matrix}∞ & for & a > 0 \\ −∞ & for & a < 0 \\ undefined & for & a = 0 \end{matrix}

• f ∈ C(I) means ”all functions which are continuous on the interval I.

• f ∈ C^{n}(I) means ”all functions which are n times continuously on differentiable on the interval I.

• a^{x} = e^{x·ln a}

• e = e^{1} = exp(1)

• ln e = 1 ⇝ e^{x} = e^{x·ln e} = e^{x·1}

• The power function follows these rules:

a^{0} = 1
a^{1} = a
a^{x+y} = a^{x} · a^{y}, x, y ∈ ℝ
(a · b)^{x} = a^{x} · b^{x}, x ∈ ℝ
a^{−x} = \frac{1}{a^{x}} , x ∈ ℝ
log(a^{x}) = x log(a), x ∈ ℝ
(a^{x})^{y} = a^{x·y} ≠ a^{x^{y}}, x, y ∈ ℝ

• Some rules for trigonometric functions:

sin^{2} x + cos^{2} x = 1
sin(x ± y) = sin(x) cos(y) ± cos(x) sin(y)
cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y)
cos(2x) = cos^{2}(x) − sin^{2}(x) = 2 cos^{2}(x) − 1
sin(2x) = 2 sin(x) cos(x)

tan x = \frac{sin x}{cos x}
cot x = \frac{cos x}{sin x}
tan x · cot x = 1
1 + tan^{2} x = \frac{1}{cos^{2} x}
1 + cot^{2} x = \frac{1}{sin^{2} x}

1.2 Special Limits

• Some special limits of sequences:

lim_{n→∞} \frac{1}{n} = 0
lim_{n→∞} ^{n} \sqrt{n} = 1
lim_{n→∞} ^{n} \sqrt{n!} = ∞
lim_{n→∞} ^{n} \sqrt{x} = 1, x > 0
lim_{n→∞} \frac{a^{n}}{n!} = 0
lim_{n→∞} (1 + \frac{x}{n})^{n} = e^{x}  ⇔  e = lim_{n→∞} (1 + \frac{1}{n})^{n} = lim_{n→∞} (\frac{1 + n}{n})^{n}
lim_{n→∞} k^{n} = \{ \begin{matrix}0 & for & |k| < 1 \\ 1 & for & k = 1 \end{matrix} The sequence (in the last example) diverges for |k| > 1

• The Euler transcendent number:

e^{x} = lim_{n→∞} (1 + \frac{x}{n})^{n} = ∑_{k=0}^{∞} \frac{x^{k}}{k!} , x ∈ ℝ

1.3 Sequences

• A sequence is called arithmetic sequence if:

a_{n+1} − a_{n} = d = const.  for ∀n ∈ ℕ ⟶ a_{n} = a_{1} + (n − 1)d  n ∈ ℕ

that is, if the difference between two elements of the sequence is an argument dependent on the first element and the product of n and a constant - hence the difference between two elements is constant.

• A sequence is called geometric sequence if:

\frac{a_{n+1}}{a_{n}} = q = const.  for ∀n ∈ ℕ ⟶ a_{n} = a_{1}q^{n−1}  n ∈ ℕ

that is, if the difference between two elements of the sequence is an argument determined by a quotient depending on n.

• We say that a sequence ⟨x_{n}⟩_{n∈ℕ} converges to a ∈ ℝ as n tends to ∞ if, for each ϵ > 0, there is a natural number N := N(ϵ), such that one has

|x_{a} − a| < ϵ, n ≥ ℕ

In this case we write lim_{n→∞} x_{n} = a • A sequence is called null sequence if it converges to zero.

• If a sequence ⟨x_{n}⟩_{n∈ℕ} converges to a ∈ ℝ then also each of its subsequences converges to a.

• A sequence ⟨x_{n}⟩_{n∈ℕ} diverges or is divergent if it is not convergent. In this case we also say that the limit lim_{n→∞} x_{n} does not exist.

• For lim_{n→∞} x_{n} = a and lim_{n→∞} y_{n} = b it holds:

1. lim_{n→∞} (x_{n} ± y_{n}) = a ± b

2. lim_{n→∞} (x_{n} · y_{n}) = a · b

3. lim_{n→∞} (αx_{n}) = α · lim_{n→∞} x_{n} = α · a, α ∈ ℝ

4. lim_{n→∞} (αx_{n}) = α · lim_{n→∞} x_{n} = α · a, α ∈ ℝ

• A sequence ⟨a_{n}⟩_{n∈ℕ} is being called limited for ∀n ∈ ℕ, if:

|a_{n}| ≤ k for k ∈ ℝ

• A sequence ⟨a_{n}⟩_{n∈ℕ} is called strictly monotonic if for all ∀n ∈ ℕ holds:

a_{n+1} > a_{n}  (strictly monotonic increasing)
a_{n+1} ≥ a_{n}  (monotonic non-decreasing)
a_{n+1} < a_{n}  (strictly monotonic decreasing)
a_{n+1} ≤ a_{n}  (monotonic non-increasing)

• If one has to calculate the limit of a sequence, the limit is the quotient of the highest powers of the

enumerator n, for the form:

a_{n} = \frac{b · n^{w} + c · n^{x}}{d · n^{y} + e · n^{z}}

Simply divide the whole term by the highest power and cancel everything tending to zero. The result will usually be a simple expression.

• If one has a sequence similar to this:

lim_{n→∞} a_{n} = (\frac{n}{n + x})^{n} → add & subtract x: (\frac{n + x − x}{n + x})^{n} = (\frac{n + x}{n + x} − \frac{x}{n + x})^{n}
= (1 − \frac{x}{n + x})^{n} which yields: (1 ± \frac{x}{n + x})^{n} = e^{±x}

• If one has something similar to (without denominator):

a_{n} = \sqrt{x} − \sqrt{y} = (\sqrt{x} − \sqrt{y}) · 1 = (\sqrt{x} − \sqrt{y}) · \frac{\sqrt{x} + \sqrt{y}}{\sqrt{x} + \sqrt{y}}

Divide or multiply with \sqrt{x} + \sqrt{y} to transform the term into the form (a − b) · (a + b) = a^{2} + b^{2}

Example assuming lim_{n→∞} a_{n}:

a_{n} = n\sqrt{n^{4} + n} − n\sqrt{n^{4} − n} ⟶ substitute: a = \sqrt{n^{4} + n} & b = \sqrt{n^{4} − n}
n · (a − b) = n · (a − b) · 1 = n · (a − b) · \frac{a + b}{a + b} = n · \frac{(a − b) · (a + b)}{a + b} = n · \frac{a^{2} + b^{2}}{a + b}
= n · (\frac{(n^{4} + n) − (n^{4} − n)}{\sqrt{n^{4} + n} − \sqrt{n^{4} − n}}) = \frac{2n^{2}}{\sqrt{n^{4} + n} − \sqrt{n^{4} − n}}
= \frac{\frac{2n^{2}}{n^{2}}}{\sqrt{\frac{n^{4}}{n^{4}} + \frac{n}{n^{4}}} − \sqrt{\frac{n^{4}}{n^{4}} − \frac{n}{n^{4}}}} = \frac{2}{\sqrt{1 + \frac{1}{n^{3}}} − \sqrt{1 − \frac{1}{n^{3}}}} = \frac{1}{\sqrt{1}} = 1

1.4 Series

For a series S_{k} = ∑_{k=1}^{∞} a_{k} one should first examine the sequence a_{k}!

• Similarly to arithmetic and geometric sequences we also have harmonic and geometric series: – For the geometric series it is even possible to determine the limit:

∑_{k=1}^{∞} x^{k} = \{ \begin{matrix}\frac{1}{1−x}, |x| < 1 \\ diverges for |x| ≥ 1 \end{matrix}

Example:

∑_{k=0}^{∞} 9 · (\frac{1}{10})^{k} = 9 · ∑_{k=0}^{∞} (\frac{1}{10})^{k} = 9 · \frac{1}{1 − \frac{1}{10}} = 9 · \frac{1}{\frac{9}{10}} = 10 · \frac{9}{9} = 10

– For the harmonic series:

∑_{k=1}^{∞} \frac{1}{k} = +∞

– There also exists the special case of the alternating harmonic series:

∑_{k=1}^{∞}(−1)^{k} \frac{1}{k} , where the Leibniz test is employed to determine a possible convergence.

– There are also the so called power series (the geometric series is a special case of it):

∑_{k=0}^{∞} g_{k}(x), x ∈ I

where the g_{k}’s are functions defined on an interval I.

• A series may converge, if the limit of the respective sequence tends to zero:

∑_{k=1}^{∞} a_{k}  converges  ⇒  lim_{k→∞} a_{k} = 0

Proof:

s_{k} := ∑_{k=1}^{∞} a_{k} ⇒ \{ \begin{matrix}lim_{k→∞} s_{k} & = & s \\ lim_{k→∞} s_{k−1} & = & s \end{matrix}

Hence: a_{k} = s_{k} − s_{k−1} ⇝ s − s = 0 (n → ∞)

This also implies that, if lim_{k→∞} a_{k} ≠ 0 or lim_{k→∞} a_{k} does not exist ⇒ ∑_{k=1}^{∞} a_{k} diverges!

• Definition: ∑_{k=1} ∞a_{k} is absolutely convergent if ∑_{k=1} ∞ |a_{k}|. Absolute convergence implies normal convergence!

• Rearrangement theorem: If a series is absolutely convergent, then each series formed by rearrangement of its terms converges to the same limit.

• The radius of convergence is determined by the following rule for ∑_{k=a}^{b} c_{k}:

R = \frac{c_{k}}{c_{k+1}}

• Convergence criteria for series of the type ∑_{k=1}^{∞} a_{k} ...

1. Comparison test: Let some k_{0} ∈ ℕ exist such that |a_{k}| ≤ b_{k}, k ≥ k_{0}. Then one has

(a) ∑_{k=1}^{∞} b_{k} converges  ⇝  ∑_{k=1}^{∞} a_{k} converges absolutely.
(b) ∑_{k=1}^{∞} |a_{k}| = 0  ⇝  ∑_{k=1}^{∞} b_{k} = 0

2. Ratio test: If a_{k} ≠ 0 is true for all k ≥ k_{0} with some k_{0} ∈ ℕ and if the sequence \frac{a_{k+1}}{a_{k}} converges properly or improperly then one has:

lim_{k→∞} \frac{a_{k+1}}{a_{k}} \{ \begin{matrix}< 1 & The series converges \\ > 1 & The series diverges \\ = 1 & Undefined \end{matrix}

Example assuming (a_{k} = \frac{x^{k}}{k!} ; a_{k+1} = \frac{x^{k+1}}{(k+1)!}):

lim_{k→∞} \frac{a_{k+1}}{a_{k}} = lim_{k→∞} \frac{x^{k+1} · k!}{(k + 1)! · x^{k}} = lim_{k→∞} \frac{x · x^{k} · k!}{k! · (k + 1) · x^{k}} = lim_{k→∞} \frac{x}{k + 1} < 1 ⟶ series converges

3. Root test: Let the sequence ( ^{k} \sqrt{|a_{k}|})_{k≥1} converge properly or improperly, then the following holds true:

lim_{k→∞} ^{k} \sqrt{|a_{k}|} \{ \begin{matrix}< 1, & The series absolutely converges \\ > 1, & The series diverges \\ = 1, & Undefined \end{matrix}

Example assuming (a_{k} = \frac{x^{k}}{k!} ):

k \sqrt{|a_{k}|} = ^{k} \sqrt{\frac{x^{k}}{k!}} = \begin{matrix}^{k} & \sqrt{x^{k}} \\ ^{k} & \sqrt{k!} \end{matrix} = \frac{x}{\sqrt[k]{k!}} = 0 < 1 ⟶ series converges

4. Leibniz test: Assume c_{k} ≥ c_{k+1} ≥ 0, k ∈ ℕ and lim_{k→∞} c_{k} = 0 then the alternating series ∑_{k=1}^{∞}(−1)^{k}c_{k} converges.

1.5 Polynomials

• Polynomials should first be checked for a zero. Afterwards one divides the polynomial P (x) through the term (x − x_{1}) where x_{1} is the first found zero of the polynomial. If the polynomial is of second degree after this, the following approach is used:

ax^{2} + bx + c  ⇝  \frac{−b ± \sqrt{b^{2} − 4ac}}{2a}

or the special case where the quadratic argument has no coefficient:

x^{2} + px + q  ⇝  \frac{−p}{2} ± \sqrt{(\frac{p}{2})^{2} − q}

• If the sum of the coefficients is zero, the first zero of the polynomial is x_{1} = 1.

For an ”ordered” polynomial P (x) = a_{k}x^{n} + a_{k−1}x^{k−1} + ... + a_{1}x + a_{0} one has a_{k} as the first ”odd” coefficient and a_{k+1} as the first ”even” coefficient. If there are coefficients which are zero, they should be written to not mix up the order of odd and even elements.

• If the sum of the ”odd” and the ”even” coefficients is equal, another zero of the polynomial is at x_{2} = −1.

1.6 Limits of Functions

• For limits of functions there hold similar relations true as for limits of sequences:

1. lim_{x→x_{0}} (f(x) ± ±g(x)) = lim_{x→x_{0}} f(x) ± lim_{x→x_{0}} g(x) = a ± b

2. lim_{x→x_{0}} (f(x) · ±g(x)) = lim_{x→x_{0}} f(x) · lim_{x→x_{0}} g(x) = a · b

3. lim_{x→x_{0}} \frac{f(x)}{g(x)} = \frac{lim_{x→x_{0}} f(x)}{lim_{x→x_{0}} g(x)} = \frac{a}{b}

• If one has a function of the form:

lim_{x→x_{0}} \frac{f(x)}{g(x)} = \frac{0}{0} or lim_{x→x_{0}} \frac{f(x)}{g(x)} = \frac{∞}{∞}

the rule of L’Hospital says that:

lim_{x→x_{0}} \frac{f(x)}{g(x)} = lim_{x→x_{0}} \frac{f′(x)}{g′(x)}

Hence if one faces such an undetermined form, derive the two functions and check the limit. In case the first (second, ...) derivative does not yield a suitable solution, just derive once more. Cannot be used with forms like \frac{a}{0} or \frac{a}{∞} because they are determined as shown in the section General assumptions.

1.7 The Derivative

• The definition of the derivative is as follows:

\frac{Δy}{Δx} = \frac{f(x + Δx) − f(x)}{Δx}  ⇝  \frac{dy}{dx} = lim_{Δx→0} \frac{f(x + Δx) − f(x)}{Δx} = f′(x)

Example f(x) = 3x^{2} + 7x − 8:

f′(x) = lim_{Δx→∞} \frac{(3(x + Δx)^{2} + 7(x + Δx) − 8) − (3x^{2} + 7x − 8)}{Δx}
= lim_{Δx→∞} \frac{(3(x^{2} + 2xΔx + Δx^{2}) + 7(x + Δx) − 8) − (3x^{2} + 7x − 8)}{Δx}
= lim_{Δx→∞} \frac{3x^{2} + 6xΔx + 3Δx^{2} + 7x + 7Δx − 8 − 3x^{2} − 7x + 8}{Δx}
= lim_{Δx→∞} \frac{6xΔx + 3Δx^{2} + 7Δx}{Δx} = lim_{Δx→∞} \frac{Δx (6x + 3Δx + 7)}{Δx}
= lim_{Δx→∞} (6x + 3Δx + 7) = 6x + 7

• Some basic derivatives (for valid ranges of definition):

f(x)  f′(x)  f′′(x)
a = const.  0  0
x^{n}  nx^{n−1}  n(n − 1)x^{n−2}
sin x  cos x  − sin x
cos x  − sin x  − cos x
tan x  \frac{1}{cos^{2} x} = 1 + tan^{2} x 2 tan x(1 + tan^{2} x)
arcsin x  \frac{1}{\sqrt{1−x^{2}}}  \frac{x}{(1−x^{2})\sqrt{1−x^{2}}}
arccos x  −\frac{1}{\sqrt{1−x^{2}}}  −\frac{x}{(1−x^{2})\sqrt{1−x^{2}}}
arctan x  \frac{1}{1+x^{2}}  \frac{−2x}{(1+x^{2})^{2}}
e^{x}  e^{x}  e^{x}
e^{c·x}  c · e^{c·x}  c^{2} · e^{c·x}
log_{a} x  \frac{1}{x·ln a}  \frac{−1}{x^{2}·ln a}
ln x  \frac{1}{x}  −\frac{1}{x^{2}}

• Mean value theorem of differentiation: Let f ∈ C^{1}[a, b], then for each pair of points x, y ∈ [a, b]with x < y some point ξ ∈ (x, y) exists such that

f(y) − f(x) = f′(ξ)(x − y)

1.8 Taylor Polynomial

P_{n}(x) = f(x_{0}) + \frac{f′(x_{0})}{1!}(x − x_{0}) + \frac{f′′(x_{0})}{2!}(x − x_{0})^{2} + ... + \frac{f^{(}n)(x_{0})}{n!}(x − x_{0})^{n} + R_{n}(x)

References

[1] Reemtsen Rempbert Prof. Dr.

Mathematics II - Calculus. 2004 BTU Cottbus.

[2] Meyberg Kurt, Vachenauer Peter.

Höhere Mathematik 1 und 2. 1991 Springer Verlag Berlin - Heidelberg - New York.

[3] Rapsch Ilse.

Folgen - Reihen - Grenzwerte. 1998 Verlag Franzbecker, Hildesheim, Berlin.

[4] Precht Dr. Manfred, Voit Dipl.-Math. Karl, Kraft Dipl.-Ing. agr. Roland. Mathematik 2 für Nichtmathematiker. 1991 R. Oldenbourg Verlag München Wien.

[5] Scheid Prof. Dr. Harald (Bearb.), Engesser Hermann (Hrsg.).

Duden Rechnen und Mathematik: das Lexikon für Schule und Praxis. 1994 Bibliographisches Institut & F.A. Brockhaus AG, Mannheim.