25 Integral Test for Series Convergence, Stirling's Formula to Approximate Factorial, Summary

Lecture from: 29.05.2024 | Video: Video ETHZ

Clicker Question: A Famous Improper Integral

What is the value of ${\int }_{-\infty }^{\infty }\frac{1}{1+x^2}dx$? Answer: $\pi$

Reasoning

Strategy: Split, Evaluate, Combine! To tackle an integral over the entire real line $(-\infty ,\infty )$, we first need to check if it “behaves” at both ends. The standard way is to split it into two pieces at some convenient point (like $x=0$), and then evaluate each piece as an improper integral. If both pieces converge to finite values, then the original integral converges to their sum.

An antiderivative (“Stammfunktion”) of $\frac{1}{1+x^2}$ is $\arctan (x)$.

Let’s evaluate the two halves:

1. Integral from $0$ to $\infty$: ${\int }_0^{\infty }\frac{1}{1+x^2}dx={\lim }_{b\to \infty }{\int }_0^b\frac{1}{1+x^2}dx$ $={\lim }_{b\to \infty }[\arctan (x){]}_0^b={\lim }_{b\to \infty }(\arctan (b)-\arctan (0))$ We know $\arctan (0)=0$. As $b\to \infty$, $\arctan (b)\to \frac{\pi }{2}$ (the arctangent function approaches its horizontal asymptote). So, ${\int }_0^{\infty }\frac{1}{1+x^2}dx=\frac{\pi }{2}-0=\frac{\pi }{2}$. This part converges.

2. Integral from $-\infty$ to $0$: ${\int }_{-\infty }^0\frac{1}{1+x^2}dx={\lim }_{a\to -\infty }{\int }_a^0\frac{1}{1+x^2}dx$ $={\lim }_{a\to -\infty }[\arctan (x){]}_a^0={\lim }_{a\to -\infty }(\arctan (0)-\arctan (a))$ As $a\to -\infty$, $\arctan (a)\to -\frac{\pi }{2}$. So, ${\int }_{-\infty }^0\frac{1}{1+x^2}dx=0-(-\frac{\pi }{2})=\frac{\pi }{2}$. This part also converges.

Combining the pieces: According to our definition for improper integrals that are open/unbounded on both sides, we pick an arbitrary point $c\in \mathbb{R}$ (let’s use $c=0$ here, as we effectively did above) and sum the two improper integrals: ${\int }_{-\infty }^{\infty }\frac{1}{1+x^2}dx={\int }_{-\infty }^c\frac{1}{1+x^2}dx+{\int }_c^{\infty }\frac{1}{1+x^2}dx$ Using $c=0$: ${\int }_{-\infty }^{\infty }\frac{1}{1+x^2}dx=\frac{\pi }{2}+\frac{\pi }{2}=\pi$ Since both parts converge, the whole integral converges to $\pi$. This is a well-known and beautiful result!

The Integral Test for Series Convergence

Connecting Infinite Sums and Infinite Areas This theorem, often called the Integral Test, provides a powerful bridge between the convergence of an infinite series $\sum f(n)$ and the convergence of an improper integral ${\int }_1^{\infty }f(x)dx$. It applies to positive, decreasing functions. The intuition is that the sum can be seen as an approximation of the area under the curve (using rectangles of width 1), and vice-versa.

Theorem: The Integral Test

Let $f:[1,\infty )\to [0,\infty )$ be a monotonically decreasing function (so $f(x)\ge 0$ and $f(x_1)\ge f(x_2)$ if $x_1<x_2$).

Then the infinite series ${\sum }_{n=1}^{\infty }f(n)$ converges if and only if the improper integral ${\int }_1^{\infty }f(x)dx$ converges.

Proof Idea (Using “Staircase” Functions)

The core idea is to compare the series with the integral by “sandwiching” the function $f(x)$ between two simpler, “step-like” functions built from the series terms.

Let $\lfloor x\rfloor$ be the floor function (the greatest integer less than or equal to $x$). Define two auxiliary functions for $x\ge 1$:

• $g(x)=f(\lfloor x\rfloor )$ (This function is constant on intervals $[n,n+1)$, taking the value $f(n)$).
• $h(x)=f(\lfloor x\rfloor +1)$ (This function is constant on intervals $[n,n+1)$, taking the value $f(n+1)$).

In the image:

• On the interval $[1,2)$, $g(x)=f(\lfloor x\rfloor )=f(1)$. This forms an “upper” rectangle.
• On the interval $[1,2)$, $h(x)=f(\lfloor x\rfloor +1)=f(1+1)=f(2)$. This forms a “lower” rectangle.

Since $f$ is monotonically decreasing: For any $x\in [n,n+1)$, we have $n\le x<n+1$. Because $f$ is decreasing, $f(n+1)\le f(x)\le f(n)$. This translates to: $h(x)\le f(x)\le g(x)$ for all $x\in [1,\infty )$.

Now, let’s integrate these functions from $1$ to some large integer $N$: ${\int }_1^{N}h(x)dx\le {\int }_1^{N}f(x)dx\le {\int }_1^{N}g(x)dx$.

Let’s evaluate the integrals of $g(x)$ and $h(x)$:

• ${\int }_1^{N}g(x)dx={\int }_1^{N}f(\lfloor x\rfloor )dx={\sum }_{n=1}^{N-1}{\int }_n^{n+1}f(n)dx={\sum }_{n=1}^{N-1}f(n)\cdot ((n+1)-n)={\sum }_{n=1}^{N-1}f(n)$. (The integral of $g(x)$ from $1$ to $N$ is the sum of the areas of the “upper” rectangles, which corresponds to the sum of the series terms $f(1)+f(2)+\cdots +f(N-1)$).
• ${\int }_1^{N}h(x)dx={\int }_1^{N}f(\lfloor x\rfloor +1)dx={\sum }_{n=1}^{N-1}{\int }_n^{n+1}f(n+1)dx={\sum }_{n=1}^{N-1}f(n+1)\cdot 1={\sum }_{n=2}^{N}f(n)$. (The integral of $h(x)$ from $1$ to $N$ is the sum of the areas of the “lower” rectangles, corresponding to $f(2)+f(3)+\cdots +f(N)$).

So, we have the crucial chain of inequalities: ${\sum }_{n=2}^{N}f(n)\le {\int }_1^{N}f(x)dx\le {\sum }_{n=1}^{N-1}f(n)$

Now, we consider the two directions of the “if and only if”

1. If ${\sum }_{n=1}^{\infty }f(n)$ converges: This means the sequence of partial sums $S_N={\sum }_{n=1}^{N}f(n)$ is bounded above. From the right inequality, ${\int }_1^{N}f(x)dx\le {\sum }_{n=1}^{N-1}f(n)\le {\sum }_{n=1}^{\infty }f(n)$ (which is a finite number). The function $F(N)={\int }_1^{N}f(x)dx$ is monotonically increasing (since $f(x)\ge 0$). Since $F(N)$ is monotonically increasing and bounded above, ${\lim }_{N\to \infty }F(N)={\int }_1^{\infty }f(x)dx$ must converge.

2. If ${\int }_1^{\infty }f(x)dx$ converges: This means ${\lim }_{N\to \infty }{\int }_1^{N}f(x)dx$ is a finite number. From the left inequality, ${\sum }_{n=2}^{N}f(n)\le {\int }_1^{N}f(x)dx\le {\int }_1^{\infty }f(x)dx$ (which is finite). So, the partial sums ${\sum }_{n=2}^{N}f(n)$ are bounded above. Since $f(n)\ge 0$, the series ${\sum }_{n=2}^{\infty }f(n)$ is a series of non-negative terms with bounded partial sums, so it must converge. If ${\sum }_{n=2}^{\infty }f(n)$ converges, then ${\sum }_{n=1}^{\infty }f(n)=f(1)+{\sum }_{n=2}^{\infty }f(n)$ also converges.

This completes the argument.

Example: The p-Series ${\sum }_{n=1}^{\infty }\frac{1}{n^{\alpha }}$

When does the series ${\sum }_{n=1}^{\infty }\frac{1}{n^{\alpha }}$ converge (for $\alpha >0$)? The function $f(x)=\frac{1}{x^{\alpha }}$ is positive and monotonically decreasing for $x\ge 1$ if $\alpha >0$. So we can apply the Integral Test. We need to check the convergence of the corresponding improper integral ${\int }_1^{\infty }\frac{1}{x^{\alpha }}dx$.

We’ve already answered this question! (In Example 5.8.2 (2) from the previous lecture notes). The integral ${\int }_1^{\infty }\frac{1}{x^{\alpha }}dx$ converges if and only if $\alpha >1$.

Therefore, by the Integral Test, the series ${\sum }_{n=1}^{\infty }\frac{1}{n^{\alpha }}$ converges if and only if $\alpha >1$. This is a very important result for series convergence!

Example: Series involving $\ln (n)$

When does the series ${\sum }_{n=2}^{\infty }\frac{1}{n(\ln n{)}^{\beta }}$ converge? (We start at $n=2$ because $\ln (1)=0$). The function $f(x)=\frac{1}{x(\ln x{)}^{\beta }}$ is positive and decreasing for $x\ge 2$ (if $\beta \ge 0$, or if $\beta <0$ and $x$ is large enough). We would compare this to the integral ${\int }_2^{\infty }\frac{1}{x(\ln x{)}^{\beta }}dx$. Let $u=\ln x$, so $du=\frac{1}{x}dx$. The integral becomes ${\int }_{\ln 2}^{\infty }\frac{1}{u^{\beta }}du$. From our previous p-series integral result (replacing $x$ with $u$ and $\alpha$ with $\beta$), this integral converges if and only if $\beta >1$. So, the series ${\sum }_{n=2}^{\infty }\frac{1}{n(\ln n{)}^{\beta }}$ converges if and only if $\beta >1$. (See script for more details on checking the conditions for $f(x)$).

Conclusion: Stirling’s Formula - Approximating Factorials

A Remarkable Approximation for $n!$ Factorials $n!=1\cdot 2\cdot \cdots \cdot n$ grow incredibly fast. Stirling’s formula gives an amazing approximation for $n!$ when $n$ is large, connecting it to powers of $n$, $e$, and $\pi$. It often arises from approximating $\ln (n!)=\sum \ln (k)$ with $\int \ln (x)dx$.

Definition: Asymptotic Equivalence

Two sequences of positive real numbers, $(a_n{)}_{n\in \mathbb{N}}$ and $(b_n{)}_{n\in \mathbb{N}}$, are called asymptotically equivalent if ${\lim }_{n\to \infty }\frac{a_n}{b_n}=1$ Notation: $a_n\sim b_n(\text{as }n\to \infty )$. This means that for large $n$, $a_n$ is very close to $b_n$ in a relative sense.

Theorem: Stirling’s Formula (Asymptotic Version)

$n!\sim \sqrt{2\pi n}\cdot n^n\cdot e^{-n}(\text{as }n\to \infty )$ This means $n!\approx \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n$ for large $n$.

Proof Idea (Sketch using Integral Approximation of $\ln (n!)$)

The proof is quite involved and often uses techniques like approximating sums by integrals, or more advanced methods. A common starting point is to consider $\ln (n!)$: $\ln (n!)=\ln (1)+\ln (2)+\cdots +\ln (n)={\sum }_{k=1}^n\ln (k)$.

This sum can be approximated by the integral ${\int }_1^n\ln (x)dx$. We know $\int \ln (x)dx=x\ln (x)-x$. So, ${\int }_1^n\ln (x)dx=[x\ln (x)-x{]}_1^n=(n\ln (n)-n)-(1\ln (1)-1)=n\ln (n)-n+1$.

The sum ${\sum }_{k=1}^n\ln (k)$ can be more precisely related to ${\int }_1^n\ln (x)dx$ using the trapezoidal rule for approximating integrals, or by carefully comparing the sum to areas of rectangles above and below the $\ln (x)$ curve.

The area of the trapezoid under the graph of $\ln (x)$ from $x$ to $x+1$ is $\frac{1}{2}(\ln (x)+\ln (x+1))$.

Summing these trapezoids from $x=1$ to $n-1$: ${\text{Area}}_{\text{trapezoids}}={\sum }_{k=1}^{n-1}\frac{1}{2}(\ln (k)+\ln (k+1))$ $=\frac{1}{2}(\ln (1)+\ln (2))+\frac{1}{2}(\ln (2)+\ln (3))+\cdots +\frac{1}{2}(\ln (n-1)+\ln (n))$ $=\frac{1}{2}\ln (1)+\ln (2)+\ln (3)+\cdots +\ln (n-1)+\frac{1}{2}\ln (n)$ $={\sum }_{k=1}^n\ln (k)-\frac{1}{2}\ln (1)-\frac{1}{2}\ln (n)=\ln (n!)-\frac{1}{2}\ln (n)$ (since $\ln (1)=0$).

Let $c_n={\int }_1^n\ln (x)dx-\left(\ln (n!)-\frac{1}{2}\ln (n)\right)$. $c_n=(n\ln (n)-n+1)-\ln (n!)+\frac{1}{2}\ln (n)=(n+\frac{1}{2})\ln (n)-n+1-\ln (n!)$. It can be shown that this sequence $c_n$ (which represents the accumulated difference between the integral and the trapezoidal sum) converges to a constant $C$ as $n\to \infty$.

The proof involves showing $c_n$ is non-negative and monotonically increasing (or decreasing, depending on the precise setup) and bounded. This means $\ln (n!)\approx (n+\frac{1}{2})\ln (n)-n+1-C$.

Exponentiating this gives $n!\approx e^{(n+1/2)\ln (n)-n+1-C}=n^{n+1/2}{e}^{-n}{e}^{1-C}=\sqrt{n}\cdot n^n{e}^{-n}\cdot (\text{some constant }A)$. The constant $A$ turns out to be $\sqrt{2\pi }$. Determining this exact constant usually requires a more sophisticated argument, often involving Wallis’s product for $\pi /2$: $\frac{\pi }{2}={\lim }_{n\to \infty }\frac{(2^{n}n!{)}^4}{((2n)!{)}^2(2n+1)}={\lim }_{n\to \infty }\frac{2^{4n}(n!{)}^4}{((2n)!{)}^2(2n+1)}$ By substituting the asymptotic form $n!\sim A\sqrt{n}{n}^n{e}^{-n}$ into Wallis’s product, one can solve for $A$ and find $A=\sqrt{2\pi }$.

This leads to the full Stirling’s formula: $n!\sim \sqrt{2\pi n}{n}^n{e}^{-n}$.

Summary of the Course (Chapters 1-5)

Let’s take a moment to look back at the major themes and results from the first five chapters. This helps consolidate what we’ve learned and see how the ideas connect.

Chapter 1: Real Numbers, Euclidean Spaces, Complex Numbers

• Real Numbers ($\mathbb{R}$): Field axioms (how addition/multiplication work), order axioms (how comparison works).
• Key Difference between $\mathbb{Q}$ and $\mathbb{R}$: $\mathbb{R}$ is order complete (or has the least upper bound property). This is fundamental!
• Consequences of Completeness: Existence of square roots (e.g., $\sqrt{2}$ is real but not rational), supremum and infimum for bounded sets.
• ${\mathbb{R}}^n$ as a Real Vector Space: Vectors, addition, scalar multiplication.
• Scalar (Dot) Product in ${\mathbb{R}}^n$, Cross Product in ${\mathbb{R}}^3$.
• Complex Numbers ($\mathbb{C}$): Definition $z=a+ib$, operations, polar form $z=r{e}^{i\theta }$.
• Roots of Unity (Kreisteilungsgleichung), Fundamental Theorem of Algebra (“All polynomials over $\mathbb{C}$ can be fully factored into linear terms”).

Chapter 2: Sequences and Series

• Sequences: Definition, convergence, limits.
• Limit Laws for sequences (sum, product, quotient).
• Weierstrass Theorem for Monotonic Sequences: “Monotonic bounded sequences converge.”
• Limes Superior, Limes Inferior: Greatest/smallest accumulation points of a sequence.
• Cauchy Sequences: Sequences whose terms eventually get arbitrarily close to each other. (In $\mathbb{R}$ and $\mathbb{C}$, Cauchy $⟺$ convergent).
• Nested Interval Theorem.
• Bolzano-Weierstrass Theorem: “Bounded sequences (in ${\mathbb{R}}^n$) have convergent subsequences.”
• Sequences in ${\mathbb{R}}^n$ and $\mathbb{C}$.
• Series:
  • Convergence of series (as limits of partial sums).
  • Geometric series.
  • Comparison tests.
  • Absolute Convergence ($\sum |a_n|$ converges) $⟹$ convergence of $\sum a_n$.
  • Alternating series, Leibniz criterion.
  • Rearrangements of series (absolutely convergent series can be rearranged freely).
  • Ratio and Root tests for convergence.
  • Power Series and their radius of convergence (absolute convergence inside, divergence outside).
  • Double series, Cauchy product of series.
  • Exponential Series $\exp (z)=\sum \frac{z^k}{k!}$ and its addition theorem $\exp (z+w)=\exp (z)\exp (w)$.

Chapter 3: Continuous Functions

• Real-valued functions: Domain, codomain, boundedness, monotonicity.
• Continuity:
  • $ϵ-\delta$ definition.
  • Sequential continuity: ”$\lim a_n=x_0⟹\lim f(a_n)=f(x_0)$.”
• Intermediate Value Theorem: “Continuous functions on intervals take on every value between any two function values.”
• Min-Max Theorem (Extreme Value Theorem): “Continuous functions on compact (closed and bounded) intervals attain their maximum and minimum values.”
• Continuity of Inverse Functions (especially for monotonic functions).
• Real Exponential Function ($\exp :\mathbb{R}\to (0,\infty )$) and its Properties.
• Natural Logarithm ($\ln :(0,\infty )\to \mathbb{R}$) as its inverse.
• Convergence of Function Sequences:
  • Pointwise vs. Uniform convergence.
  • Continuity of the limit function under uniform convergence.
• Continuity of Power Series (within their radius of convergence).
• Trigonometric Functions ($\sin ,\cos ,\tan ,\dots$), the number $\pi$.
• Limits of Functions:
  • Definition via sequence convergence.
  • Limit laws, examples.
  • One-sided limits (${\lim }_{x\to x_0^{+}},{\lim }_{x\to x_0^{-}}$).

Chapter 4: Differentiable Functions

• Definition via limit of difference quotients: $f^{\prime }(x_0)={\lim }_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$.
• Tangent lines to functions.
• Differentiation Rules: Linearity, Product Rule, Quotient Rule, Chain Rule.
• Derivative of Inverse Functions: $(f^{-1}{)}^{\prime }(y_0)=1/f^{\prime }(x_0)$.
• Necessary Condition for Local Extrema: $f^{\prime }(x_0)=0$ (for interior points).
• Rolle’s Theorem.
• Mean Value Theorem (MVT): ”$\exists \xi \in (a,b)$ such that $f^{\prime }(\xi )=\frac{f(b)-f(a)}{b-a}$.”
• Connection between Monotonicity and the First Derivative.
• Arcus Functions (inverse trig) and their derivatives.
• L’Hôpital’s Rule for limits of "$\frac{0}{0}$" or "$\frac{\infty }{\infty }$" forms (“differentiate numerator and denominator”).
• Convex Functions.
• Higher Order Derivatives and smooth functions.
• Differentiation of Power Series (term by term).
• Taylor Approximation / Taylor Polynomials: $T_n(f;x;x_0)={\sum }_{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0{)}^k$.
• Taylor’s Theorem with Remainder: $f(x)=T_n(f;x;x_0)+\frac{f^{(n+1)}(\xi )}{(n+1)!}(x-x_0{)}^{n+1}$.
• Sufficient Criterion for Local Extrema (Higher Derivative Test: if $f^{\prime }(x_0)=\cdots =f^{(n-1)}(x_0)=0$ and $f^{(n)}(x_0)\ne 0$, then $n$ odd $⟹$ no extremum; $n$ even and $f^{(n)}(x_0)>0⟹$ local min; $n$ even and $f^{(n)}(x_0)<0⟹$ local max).

Chapter 5: The Riemann Integral

• Definition via partitions, lower/upper Darboux sums.
• Mesh size of partitions.
• Riemann Sums: $\sum f({\xi }_i)(x_i-x_{i-1})$, with ${\xi }_i\in [x_{i-1},x_i]$.
• Integrable Functions:
  • Fundamental operations preserve integrability ($f+g,\lambda f,fg,|f|$ etc.).
  • Linearity of the integral.
  • Continuous functions are integrable.
  • Monotonic functions are integrable.
• Monotonicity and Triangle Inequality for Integrals.
• Cauchy-Schwarz Inequality for Integrals: $|\int fg|\le \sqrt{\int f^2}\sqrt{\int g^2}$.
• Mean Value Theorem for Integrals.
• Fundamental Theorem of Calculus (Hauptsatz):
  • Existence of antiderivatives for continuous functions: $\frac{d}{dx}{\int }_a^{x}f(t)dt=f(x)$.
  • Calculation of definite integrals using antiderivatives: ${\int }_a^{b}f(x)dx=F(b)-F(a)$.
• Integration by Parts.
• Substitution Rule.
• Area of Circles, Volume of Spheres.
• Indefinite Integrals, Techniques of Integration.
• Partial Fraction Decomposition (for rational functions: case “all real roots in denominator,” case “also complex roots in denominator”).
• Function Sequences and Integration (uniform convergence allows swapping $\lim$ and $\int$).
• Integration of Power Series (term by term).
• Improper Integrals (unbounded intervals or functions).
• Comparison Test between Series and Improper Integrals (Integral Test).
• Stirling’s Formula: $n!\sim \sqrt{2\pi n}{n}^n{e}^{-n}$.