Numerical Series Notes

CH1: Numerical Series

Objectives

• Calculate sums in simple or classic cases.
• Study the nature of a numerical series using different criteria:
  • Series with positive terms.
  • Riemann's criterion.
  • d'Alembert's rule.
  • Cauchy's rule.

Introductory Activity: Fourier Series

• A particular form of numerical series used to decompose a periodic function into an infinite sum of trigonometric terms.
• Essential in various fields such as signal processing and data compression.
• Data compression is an essential technique in signal processing and modern communication systems.
• It reduces the amount of data needed to represent a signal while retaining essential information.
• Compressed signals include images, videos, and sounds.
• Compression:
  • Decreases the memory space required to store data.
  • Accelerates data processing and transfer while maintaining high quality.

Question

• Why are compressed audio or video files (MP3, MPEG) smaller than the originals while remaining very clear?

Answer

• Any complex signal (e.g., a sound wave or an image) can be represented as a sum of simple sinusoidal waves, thanks to Fourier series.

Example

• Let the signal be a square wave (e.g., a binary 0/1 signal) periodic with period $T = 2\pi$.
  $S(t) = \begin{cases} 1 & \text{if } 0 \leq t \leq T/2 \ 0 & \text{if } T/2 \leq t \leq T \end{cases}$
• Can be decomposed into a Fourier series (numerical series):
  $S(t) = \frac{4}{\pi} \sum_{n=1}^{\infty} \frac{1}{n} \sin(nt)$, with n is odd.
• In compression, only the most important sinusoids are retained, which reduces the amount of data.
• The convergence of Fourier series guarantees that the infinite sum is well-defined and can be used to reconstruct a signal.

Generalities on Numerical Series

Definition

• Let $n<em>0 \in \mathbb{N}$ and $(U</em>n)<em>{n \geq n</em>0}$ be a real sequence.
  • For $n \geq n<em>0$, $S</em>n = \sum<em>{k=n</em>0}^{n} U<em>k$ is called the partial sum of $(U</em>n)<em>{n \geq n</em>0}$ of order n.
  • The sequence of partial sums $(S<em>n)</em>{n \geq n<em>0}$, denoted $\sum</em>{n \geq n<em>0} U</em>n$, is called the numerical series of general term $U_n$.

Example 1

1. The partial sum of the series with general term $U<em>n = \frac{1}{n^2}$ for $n \geq 1$ is defined by: $S</em>n = \sum_{k=1}^{n} \frac{1}{k^2}$
2. The partial sum of the series with general term $U<em>n = \frac{1}{\sqrt{(n - 2)(n - 1)(n - 3)}}$ for $n \geq 4$ is defined by: $S</em>n = \sum_{k=4}^{n} \frac{1}{\sqrt{(k - 2)(k - 1)(k - 3)}}$

Convergence

• Determining the nature of a series means determining whether it is convergent or divergent.
• The convergence of the general term $(U<em>n)</em>{n \geq n<em>0}$ does not generally imply the convergence of the numerical series $\sum</em>{n \geq n<em>0} U</em>n$.
• In particular, studying the convergence of $\sum<em>{n \geq n</em>0} U<em>n$ consists of studying the convergence of $(S</em>n)<em>{n \geq n</em>0}$ and not of $(U<em>n)</em>{n \geq n_0}$.

Example 2

• Express $S<em>n$ as a function of n, then calculate $\lim</em>{n \to +\infty} S_n$.

  1. $S<em>n = \sum</em>{k=1}^{n} \frac{1}{k(k + 1)}$
  2. $S<em>n = \sum</em>{k=0}^{n} \frac{1}{2^k}$
  3. $S<em>n = \sum</em>{k=1}^{n} \ln(1 + \frac{1}{k})$
  4. $S<em>n = \sum</em>{k=1}^{n} e^{-k}$

• If $\lim<em>{n \to +\infty} S</em>n$ exists and is finite, we say that the sequence $(S_n)$ is convergent.

• If $\lim<em>{n \to +\infty} S</em>n$ is infinite or does not exist, we say that the sequence $(S_n)$ is divergent.

• The series $\sum<em>{n \geq n</em>0} U<em>n$ converges to S if the sequence of partial sums $S</em>n = \sum<em>{k=n</em>0}^{n} U<em>k$ converges to S. In this case, the limit S is called the sum of the series, and we write $\sum</em>{k=n<em>0}^{\infty} U</em>k = \lim<em>{n \to \infty} S</em>n = S$

• The notations $\sum<em>{n \geq n</em>0} U<em>n$ and $\sum</em>{n=n<em>0}^{+\infty} U</em>n$ designate two different notions:

  • The notation $\sum<em>{n \geq n</em>0} U_n$ designates a numerical series.
  • In the case where it converges, the notation $\sum<em>{n=n</em>0}^{+\infty} U_n$ designates its limit.

Remarque 4 (Attention!)

• Let $\sum<em>{n \geq n</em>0} U<em>n$ be a numerical series converging to $S = \sum</em>{n=n<em>0}^{+\infty} U</em>n$. We call the remainder of order n of this series the real number $R<em>n$ defined by: $R</em>n = S - S<em>n = \sum</em>{k=n<em>0}^{+\infty} U</em>k - \sum<em>{k=n</em>0}^{n} U<em>k = \sum</em>{k=n+1}^{+\infty} U_k$

Exercice 1

• Let $\sum<em>{n \geq 0} U</em>n$ be a numerical series converging to $S = \sum<em>{n=n</em>0}^{+\infty} U_n$.

  1. For all $n \geq 1$, calculate $R<em>{n-1} - R</em>n$.
  2. Calculate the limit of $R_n$ as n tends to infinity.
  3. Conclude.

Necessary Condition for Convergence

• To study the convergence of a numerical series $\sum<em>{n \geq n</em>0} U<em>n$, it is necessary that its general term $U</em>n$ satisfies a certain condition that we define in what follows.

Activity 1

• Let $\sum<em>{n \geq n</em>0} U<em>n$ be a convergent numerical series, and $S</em>n = \sum<em>{k=n</em>0}^{n} U_k$ its partial sum of order n.

  1. Verify that $U<em>n = S</em>n - S_{n-1}$.
  2. Calculate $\lim<em>{n \to +\infty} U</em>n$ and conclude.

• If the series $\sum<em>{n \geq n</em>0} U<em>n$ is convergent then $(U</em>n)<em>{n \geq n</em>0}$ is convergent and $\lim<em>{n \to +\infty} U</em>n = 0$.

Exercice 2

• Show that the following series are divergent.

  1. $\sum_{n \geq 0} \frac{n^2 - 1}{n^2 + 1}$
  2. $\sum_{n \geq 0} \arctan(n)$
  3. $\sum_{n \geq 0} \cos(n\pi)$

• A numerical series whose general term does not tend towards 0 is divergent.

• If $\lim<em>{n \to +\infty} U</em>n = 0$, is the series $\sum<em>{n \geq n</em>0} U_n$ convergent?

Exercice 3

• For all $n \geq 1$, we set $U<em>n = \ln(1 + \frac{1}{n})$ and $S</em>n = \sum<em>{k=1}^{n} U</em>k$.

  1. Calculate $\lim<em>{n \to +\infty} U</em>n$.
  2. Verify that $\ln(1 + \frac{1}{k}) = \ln(k + 1) - \ln(k)$, for $k \geq 1$.
  3. Calculate $S_n$.
  4. Calculate $\lim<em>{n \to +\infty} S</em>n$ and deduce the nature of the series $\sum<em>{n \geq 1} U</em>n$.
  5. What can we conclude?

Algebraic Operations on Series

• Let $\sum<em>{n \geq n</em>0} U<em>n$ and $\sum</em>{n \geq n<em>0} V</em>n$ be two numerical series, and $\lambda \in \mathbb{R}$.

Example 3

• Study the nature of each of the following numerical series:

  1. $\sum_{n \geq 2} \left(\frac{1}{n(n - 1)} + \frac{1}{2n}\right)$
  2. $\sum_{n \geq 2} \left(\frac{1}{n(n - 1)} + \arctan(n)\right)$
  3. $\sum_{n \geq 2} \left(\frac{2n + 1}{n} - \frac{2n - 1}{n - 1}\right)$

Reference Series

• In this section, we will present some reference series.

Geometric Series

Activity 2

• Let $q \in \mathbb{R}$, $n<em>0 \in \mathbb{N}$ and $S</em>n = \sum<em>{k=n</em>0}^{n} q^k$.

  1. Calculate $S<em>n$ for all $n \geq n</em>0$.
  2. Study, according to the values of q, the limit of $S_n$ as n tends towards $+\infty$.

• The geometric series $\sum<em>{n \geq n</em>0} q^n$ is convergent if and only if $q \in ]-1, 1[$ and in this case, $\sum<em>{n=n</em>0}^{+\infty} q^n = \frac{q^{n_0}}{1 - q}$.

Example 4

• Study the nature of each of the following series:

  1. $\sum_{n \geq 0} \left(\frac{9}{10}\right)^n$
  2. $\sum_{n \geq 0} 5^n$
  3. $\sum_{n \geq 0} \left(\frac{-1}{3}\right)^n$

Telescopic Series

• Let $(a<em>n)</em>{n \geq n<em>0}$ be a sequence. The series with general term $(a</em>{n+1} - a_n)$ is called a telescopic series

Activity 3

• Let $n<em>0 \in \mathbb{N}$ and $(a</em>n)<em>{n \geq n</em>0}$ be a real sequence.

  1. Express $\sum<em>{k=n</em>0}^{n} (a<em>{k+1} - a</em>k)$ as a function of $a<em>{n+1}$ and $a</em>{n_0}$.
  2. Show then that the series $\sum<em>{n \geq n</em>0} (a<em>{n+1} - a</em>n)$ and the sequence $(a<em>n)</em>{n \geq n_0}$ have the same nature.

• The telescopic series $\sum<em>{n \geq n</em>0} (a<em>{n+1} - a</em>n)$ and the sequence $(a<em>n)</em>{n \geq n_0}$ are of the same nature.

Exercice 4

• Study the nature of each of the following series:

  1. $\sum_{n \geq 2} \frac{1}{(n - 1)n}$
  2. $\sum_{n \geq 1} \ln(1 - \frac{1}{n + 1})$

Riemann Series

• For $\alpha \in \mathbb{R}$ and $n<em>0 \in \mathbb{N}^*$, a Riemann series is a series defined by $\sum</em>{n \geq n_0} \frac{1}{n^\alpha}$.

Activity 4

• We seek to determine a convergence criterion for Riemann series.

  1. Suppose that $\alpha \leq 0$, calculate $\lim<em>{n \to +\infty} \frac{1}{n^\alpha}$ and deduce that the series $\sum</em>{n \geq 1} \frac{1}{n^\alpha}$ is divergent.
  2. Suppose that $\alpha > 0$. Let $k \geq 1$ be an integer, noting that the function $t \mapsto \frac{1}{t^\alpha}$ is decreasing on $[k, k + 1]$, show that
     $\frac{1}{(k + 1)^\alpha} \leq \frac{1}{t^\alpha} \leq \frac{1}{k^\alpha} \quad \forall t \in [k, k + 1]$
  3. By integrating the preceding inequality between k and k + 1, then summing from k = 1 to k = n, show that
     $\sum<em>{k=2}^{n+1} \frac{1}{k^\alpha} \leq \int</em>{1}^{n+1} \frac{dt}{t^\alpha} \leq \sum_{k=1}^{n} \frac{1}{k^\alpha}$
  4. Calculate $\int_{1}^{n+1} \frac{dt}{t^\alpha}$ as a function of $\alpha$.
  5. Discuss, according to the values of $\alpha$, the nature of the Riemann series $\sum_{n \geq 1} \frac{1}{n^\alpha}$.

• The Riemann series $\sum_{n \geq 1} \frac{1}{n^\alpha}$ is convergent if and only if $\alpha > 1$.

Exercice 5

• Study the nature of the following numerical series:

  1. $\sum_{n \geq 1} \frac{1}{\sqrt{n}}$
  2. $\sum_{n \geq 1} \frac{1}{n\sqrt{n^3}}$

Bertrand Series

• A Bertrand series is a series defined by $\sum_{n \geq 2} \frac{1}{n^\alpha \cdot (\ln(n))^\beta}$ where $(\alpha, \beta) \in \mathbb{R}^2$.

Activity 5 (an idea of proof)

1. Suppose that $\alpha < 1$. We have $\lim_{n \to +\infty} n \times \frac{1}{n^\alpha \cdot (\ln(n))^\beta} = +\infty$, show that the Bertrand series is divergent.

2. Suppose that $\alpha > 1$. We have $\lim_{n \to +\infty} n^r \times \frac{1}{n^\alpha \cdot (\ln(n))^\beta} = 0$ where $1 < r < \alpha$, show that the Bertrand series is convergent.

3. Suppose that $\alpha = 1, \beta \leq 0$. Show that for all $n \geq 3$ we have: $\frac{1}{n} \leq \frac{1}{n \cdot (\ln(n))^\beta}$, and deduce that the Bertrand series is divergent.

4. Suppose that $\alpha = 1$ and $\beta > 1$.

   a) Show that for all $n \geq 3$ we have: $\frac{1}{n \cdot (\ln(n))^\beta} \leq \int_{n-1}^{n} \frac{dt}{t \cdot (\ln(t))^\beta}$.

   b) Deduce that for all $N \geq 3$ we have: $\sum_{n=3}^{N} \frac{1}{n \cdot (\ln(n))^\beta} \leq \frac{1}{\beta - 1} \cdot \frac{1}{(\ln(2))^{\beta-1}}$, then conclude that the Bertrand series converges.

• Let $(\alpha, \beta) \in \mathbb{R}^2$. The Bertrand series $\sum_{n \geq 2} \frac{1}{n^\alpha \cdot (\ln(n))^\beta}$ converges if and only if:

• $\alpha > 1$, or ($\alpha = 1$ and $\beta > 1$).

Exercice 6

• Give the nature of the following numerical series:

  1. $\sum_{n \geq 2} \frac{1}{\sqrt{n} \cdot \ln(n)}$
  2. $\sum_{n \geq 2} \frac{\ln(n)}{n^2}$
  3. $\sum_{n \geq 2} \frac{1}{n \cdot (\ln(n))^2}$
  4. $\sum_{n \geq 2} \frac{\ln(n)}{n}$

Series with Positive Terms

• In this section, we group together some convergence criteria that are only valid for series with positive terms.
• A real series $\sum<em>{n \geq n</em>0} U<em>n$ is said to have positive terms if $U</em>n \geq 0, \quad \forall n \in \mathbb{N}$.

Activity 6

• Let $n<em>0 \in \mathbb{N}$, $\sum</em>{n \geq n<em>0} U</em>n$ be a series with positive terms, and $S<em>n = \sum</em>{k=n<em>0}^{n} U</em>k$ its partial sum of order n.

  1. Calculate $S<em>{n+1} - S</em>n$, for all $n \geq n_0$.
  2. Show that the sequence $(S_n)$ is increasing.
  3. Conclude.

• Let $\sum<em>{n \geq n</em>0} U<em>n$ be a series with positive terms, then, $\sum</em>{n \geq n<em>0} U</em>n$ converges if and only if the sequence of partial sums $(S_n)$ is bounded above.

Exercice 7

• Let the sequence $U<em>n = \int</em>{n-1}^{n} \frac{1}{1 + t^2} dt, \quad \forall n \in \mathbb{N}^*$.

  1. Show that $U_n \geq 0, \quad \forall n \in \mathbb{N}^*$.
  2. Calculate $S<em>n = \sum</em>{k=1}^{n} U_k$.
  3. Show that the sequence $(S_n)$ is bounded above.
  4. Deduce the nature of the series $\sum<em>{n \geq 1} U</em>n$.

Comparison Criteria

• Let $\sum<em>{n \geq n</em>0} U<em>n$ and $\sum</em>{n \geq n<em>0} V</em>n$ be two series with positive terms such that:
  $0 \leq U<em>n \leq V</em>n , \forall n \in \mathbb{N}$

• If $\sum<em>{n \geq n</em>0} V<em>n$ converges then $\sum</em>{n \geq n<em>0} U</em>n$ converges.

• If $\sum<em>{n \geq n</em>0} U<em>n$ diverges then $\sum</em>{n \geq n<em>0} V</em>n$ diverges.

Exercice 8

• Study the nature of each of the following series:

  1. $\sum_{n \geq 1} \frac{\sin^2(n)}{n^2}$
  2. $\sum_{n \geq 1} \frac{1}{n \cos^2(n)}$

• Let $\sum<em>{n \geq n</em>0} U<em>n$ and $\sum</em>{n \geq n<em>1} V</em>n$ be two series with positive terms such that $U<em>n$ and $V</em>n$ are equivalent in the neighborhood of $+\infty$:
  $U<em>n \sim</em>{+ \infty} V<em>n \iff \lim</em>{n \to +\infty} \frac{U<em>n}{V</em>n} = 1$

• Then, $\sum<em>{n \geq n</em>1} V<em>n$ and $\sum</em>{n \geq n<em>0} U</em>n$ are of the same nature.

Exercice 9

• Study the nature of the following series:

  1. $\sum_{n \geq 1} \frac{n^2 + 1}{n^5 + 7n^2 + 8}$
  2. $\sum_{n \geq 1} \ln\left(1 + \frac{1}{n^2}\right)$
  3. $\sum_{n \geq 1} \frac{1}{n(n^2 + 1)}$

Riemann's Criterion

Activity 7

• Let $\sum<em>{n \geq n</em>0} U_n$ be a series with positive terms and $\alpha \in \mathbb{R}$.

  1. Suppose that $\alpha \leq 1$ and $\lim<em>{n \to +\infty} n^\alpha U</em>n = +\infty$.

     a) Show that there exists $N<em>1 \geq n</em>0$, such that $U<em>n \geq \frac{1}{n^\alpha} , \forall n \geq N</em>1$.

     b) Deduce the nature of the series $\sum<em>{n \geq n</em>0} U_n$.

  2. Suppose that $\alpha > 1$ and $\lim<em>{n \to +\infty} n^\alpha U</em>n = 0$.

     a) Show that there exists $N<em>2 \geq n</em>0$ such that $0 \leq U<em>n \leq \frac{1}{n^\alpha} , \forall n \geq N</em>2$.

     b) Deduce the nature of the series $\sum<em>{n \geq n</em>0} U_n$.

• Let $\sum<em>{n \geq n</em>0} U_n$ be a series with positive terms.

  • If there exists $\alpha \leq 1$ such that $\lim<em>{n \to +\infty} n^\alpha U</em>n \neq 0$ then the series $\sum<em>{n \geq n</em>0} U_n$ is divergent.
  • If there exists $\alpha > 1$ such that $\lim<em>{n \to +\infty} n^\alpha U</em>n \not= \infty$ (exists and is finite) then the series $\sum<em>{n \geq 0} U</em>n$ is convergent

Exercice 10

• Study the nature of each of the following series:

  1. $\sum_{n \geq 0} ne^{-n}$
  2. $\sum_{n \geq 1} \ln \left(1 + \frac{1}{n}\right)$
  3. $\sum_{n \geq 1} \ln \left(1 + \frac{1}{n^2}\right)$

d'Alembert's Rules

• Let $\sum<em>{n \geq n</em>0} U<em>n$ be a series with strictly positive terms such that $\lim</em>{n \to +\infty} \frac{U<em>{n+1}}{U</em>n} = l$

  • If $l < 1$ then $\sum<em>{n \geq 0} U</em>n$ is convergent.
  • If $l > 1$ then $\sum<em>{n \geq 0} U</em>n$ is divergent.
  • If $l = 1$, we cannot conclude.

Example 5 (Case $\lim<em>{n \to +\infty} \frac{U</em>{n+1}}{U_n} = 1$)

• Let $U<em>n = \frac{1}{n^\alpha}$, we have $\lim</em>{n \to +\infty} \frac{U<em>{n+1}}{U</em>n} = 1$ but $\sum_{n \geq 1} \frac{1}{n^\alpha}$ converges if $\alpha > 1$ and diverges if $\alpha \leq 1$.

Exercice 11

• Study the nature of each of the following series:

  1. $\sum_{n \geq 1} \frac{n!}{n^n}$
  2. $\sum_{n \geq 2} na^n, a \in \mathbb{R}^*$

Cauchy's Rules

• Let $\sum<em>{n \geq n</em>0} U<em>n$ be a series with strictly positive terms such that $\lim</em>{n \to +\infty} \sqrt[n]{U_n} = l$

  • If $l < 1$ then $\sum<em>{n \geq n</em>0} U_n$ is convergent.
  • If $l > 1$ then $\sum<em>{n \geq n</em>0} U_n$ is divergent.
  • If $l = 1$, we cannot conclude.

Example 6 (Case $\lim<em>{n \to +\infty} \sqrt[n]{U</em>n} = 1$)

• Let $U<em>n = \frac{1}{n^\alpha}$, we have $\lim</em>{n \to +\infty} \sqrt[n]{U<em>n} = 1$ but $\sum</em>{n \geq 1} \frac{1}{n^\alpha}$ converges if $\alpha > 1$ and diverges if $\alpha \leq 1$.

Exercice 12

• Study the nature of each of the following series:

  1. $\sum_{n \geq 1} \frac{2^{n-1}}{3^{n+1}}$
  2. $\sum_{n \geq 2} \frac{1}{(\ln(n))^n}$

• Series with Negative Terms

• A real series $\sum<em>{n \geq n</em>0} U<em>n$ is said to have negative terms if $U</em>n \leq 0 , \forall n \geq n_0$.

• For the study of the convergence of this series, we can restrict ourselves to the case of the study of the convergence of a series with positive terms by setting $V<em>n = -U</em>n$.

• Thus, all the convergence criteria used for the study of the nature of a real positive series are valid in the case where the real series has terms of constant sign.

Series with Arbitrary Terms

• When the real series $\sum<em>{n \geq n</em>0} U_n$ is not with terms of constant sign, the criteria stated previously are no longer valid.
• In this case, we need other criteria to study its convergence.

Alternating Series

• A series is said to be alternating if it is of one of the following types:
  $\sum<em>{n \geq n</em>0} (-1)^n U<em>n \text{ or } \sum</em>{n \geq n<em>0} (-1)^{n+1} U</em>n$
  where $U<em>n \geq 0 \quad \forall n \geq n</em>0$

Example 7

• The following series are alternating series:
  $\sum<em>{n \geq 1} \frac{(-1)^n}{\sqrt{n}} , \sum</em>{n \geq 1} \frac{\cos(n\pi)}{n}$.

• Let $\sum<em>{n \geq n</em>0} (-1)^n U<em>n$ be an alternating series, if the sequence $(U</em>n)<em>{n \geq n</em>0}$ is decreasing and $\lim<em>{n \to +\infty} U</em>n = 0$, then, $\sum<em>{n \geq n</em>0} (-1)^n U_n$ is convergent.

• Moreover, we have $\vert R<em>n\vert = \vert \sum</em>{k=n+1}^{+\infty} (-1)^n U<em>k\vert \leq U</em>{n+1} , \forall n \geq n_0$

Exercice 13

• Study the nature of each of the following series:

  1. $\sum_{n \geq 1} \frac{(-1)^n}{\sqrt{n}}$
  2. $\sum_{n \geq 1} \frac{(-1)^n}{n^2}$
  3. $\sum_{n \geq 1} \cos(n\pi)e^{-n}$

Absolute Convergence

• To bring the study of a series with arbitrary terms back to that of a series with positive terms, we define the absolute convergence of a numerical series.
• Let $\sum<em>{n \geq n</em>0} U<em>n$ be a numerical series with arbitrary terms, we say that the series $\sum</em>{n \geq n<em>0} U</em>n$ is absolutely convergent if $\sum<em>{n \geq n</em>0} \vert U_n \vert$ is convergent.

Exercice 14

• Show that the series $\sum_{n \geq 1} \frac{\cos(n)}{n^2}$ is absolutely convergent.

• If $\sum<em>{n \geq n</em>0} U_n$ is an absolutely convergent series then it is convergent.

Exercice 15

• Let the series $\sum_{n \geq 1} \frac{(-1)^n}{\sqrt{n}}$.

  1. Show that the series $\sum_{n \geq 1} \frac{(-1)^n}{\sqrt{n}}$ is convergent.
  2. Study the absolute convergence of the series $\sum_{n \geq 1} \frac{(-1)^n}{\sqrt{n}}$.
  3. Conclude.

• The converse of the theorem is false, there exist series that are convergent and not absolutely convergent.

Semi-convergence

• Let $\sum<em>{n \geq n</em>0} U<em>n$ be a series with arbitrary terms, we say that the series $\sum</em>{n \geq n<em>0} U</em>n$ is semi-convergent if $\sum<em>{n \geq n</em>0} U<em>n$ is convergent and $\sum</em>{n \geq n<em>0} \vert U</em>n \vert$ is divergent.

Application

• A person suffering from an illness must take a daily dose of 20mg of a certain medication. Each day, the body eliminates 25% of the medication present.

• We denote $q_n$ the quantity of medication (in mg) present in the body at the end of the nth day ($n \in \mathbb{N}^*$).

  1. Show that, for all $n \in \mathbb{N}^*$, $q<em>n = 20 \sum</em>{k=1}^{n} (\frac{3}{4})^k$.
  2. The body can only support a quantity of 70mg of this medication. Knowing that she must take this medication for the rest of her days, is this person in danger?