Suite Définie par Intégrale - Calcul Intégral - 2 Bac SM - [Exercice 19]

In this video, I will be working with you to correct an assignment on the sequence defined by the tangent integral to the nth power. We will cover order and integration techniques, integration by parts, convergence studies, and calculating the limit of certain sums. This assignment is taken from the book Al Moufid, 2nd year Baccalaureate (Science and Mathematics). Remember, it is important to try to work through the exercise before seeing the answer key. ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ 🔔 Please click the LIKE button if you enjoy our content, leave a comment, and subscribe to our YouTube channel to receive our new videos. 👉Write me your questions in the comment section if you didn't understand something! 😀 💡Please also remember to SHARE with your friends!!💪💪🔥🔥 ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ You can download the exercise instructions from here: https://cutt.ly/mxdroH2 (Homework 4, Book Al Moufid) For any natural number n, we consider the integral I_n=∫_0^(π/4)▒〖〖tan〗^n⁡x □(24&dx)〗 1) Show that (I_n )_n is positive and decreasing. 2) Show that ∶(∀n∈N) 〖I_n+I〗_(n+2)=1/(n+1) 3) Show that ∶ (∀n∈N) 1/2(n+1) ≤I_n≤1/(n+1) and deduce the limit of (I_n )_n. 4) For all n∈N we set ∶ φ(n)=I_(n+4)-I_n Calculate φ(n) as a function of n. 5)a) Calculate I_0 and I_2 . 5)b) For k∈N^* , calculate the sum ∑_(p=0)^(k-1)▒φ(4p+2) as a function of I_2 and I_(4k+2) . 5)c) Deduce (lim)┬(k→+∞) ∑_(q=0)^2k▒(-1)^q/(2q+1) 6)a) Calculate I_1 and I_3 . 6)b) For k∈N^* , calculate the sum ∑_(p=0)^(k-1)▒φ(4p+1) as a function of I_1 and I_(4k+1) . 6)c) Deduce (lim)┬(k→+∞) ∑_(q=1)^2k▒(-1)^(q+1)/q 00:38 Exercise Statement 02:05 Question 1) 06:40 Question 2) 09:38 Question 3) 12:50 Question 4) 14:58 Question 5)a) 16:25 Question 5)b) 21:14 Question 5)c) 30:05 Question 6)a) 32:34 Question 6)b) 37:31 Question 6)c) ▬▬▬▬▬▬▬ SIMILAR VIDEOS ▬▬▬▬▬▬▬ Sequence Defined by Integral - Calculus - 2 Baccalaureate Mathematics - [Exercise 19]:    • Suite Définie par Intégrale - Calcul Intég...   Sequence Defined by Integral - Integration by Parts - 2 Baccalaureate Mathematics - [Exercise 18]:    • Suite Définie par Intégrale - Integration ...   Integration by Parts - Calculus - 2 Baccalaureate Mathematics - [Exercise 17]:    • Integration par Parties - Calcul Intégral ...   Integration by Change of Variable - Integration by Parts - 2 Baccalaureate Mathematics - [Exercise 16-1/2]:    • Suite Définie par Intégrale - Integration ...   Study of a Function Defined by an Integral - Calculus of Integrals - 2 Baccalaureate Mathematics - [Exercise 15]:    • Etude de Fonction Définie par une Intégral...   Riemann Integral - Sum Riemann's Calculus - Integral Calculus - 2 Baccalaureate (Science and Mathematics) - [Exercise 14]:    • Intégrale de Riemann - Somme de Riemann - ...   Integration by Parts - Sequence and Integral - Limit of a Sequence - 2 Baccalaureate (Science and Mathematics) - [Exercise 13]:    • Suite Définie par Intégrale - Integration ...   Integration by Change of Variable - Integration by Parts - 2 Baccalaureate (Science and Mathematics) - [Exercise 12]:    • Intégration par Changement de Variable - I...   Calculus of Integrals - Integrals of Trigonometric Functions - 2 Baccalaureate (Science and Mathematics) - [Exercise 11]:    • Calcul d'Intégrales - Intégrales des Fonct...   Calculus of Integrals - Integration by Parts - 2 Baccalaureate (Science and Mathematics) - [Exercise 10]:    • Calcul d'Intégrales - Integration par Part...   INTEGRAL CALCULATIONS - Integrals of Trigonometric Functions - 2 BAC SM S.ex - [Exercise 3]:    • CALCULS D’INTÉGRALES - Intégrales des Fonc...   CALCULATING THE ANTIBINDING VALUE OF A TRIGONOMETRIC FUNCTION - 2 BAC SM S.ex - [EXERCISE 3]:    • CALCUL DE PRIMITIVE D'UNE FONCTION TRIGONO...   CALCULATING THE ANTIBINDING VALUE OF A FUNCTION Containing a Square Root - 2 BAC - [EXERCISE 2]:    • CALCUL DE PRIMITIVE D'UNE FONCTION Contena...   ANTI-PRIMITIVE AND INTEGRAL CALCULATION OF A RATIONAL FRACTION - 2 BAC SM [EXERCISE 2]:    • PRIMITIVE ET CALCUL INTÉGRALE D'UNE FRACTI...   FUNCTIONS PRIMITIVES - NATIONAL EXAM 2 BAC SCIENCE MATH - [EXERCISE 1]:    • FONCTIONS PRIMITIVES - EXAMEN NATIONAL 2 B...   INTEGRAL CALCULATIONS - Integral with Cos and Sin - 2 BAC SM S.ex - [EXERCISE 1]:    • CALCULS D’INTÉGRALES - intégrale avec Cos ...   INTEGRAL CALCULATIONS - Integral with Cos and Sin - 2 BAC SM S.ex - [EXERCISE 1]:    • CALCULS D’INTÉGRALES - intégrale avec Cos ...   ▬▬▬▬▬▬▬▬▬▬SOCIAL MEDIA▬▬▬▬▬▬▬▬▬▬ Subscribe Here:    / @mathphys   Social Media:   / 536279063660093     / math-phys-112225370317476     / mathphys5   ▬ ... national ▬▬▬▬▬▬▬▬▬ Video Recap ▬▬▬▬▬▬▬▬▬ • We will apply order relations and integration to determine the limit of the sequence In. • We will proceed with integration to demonstrate a recurrence relation with In. • We will express the integral In as a sum. • We will deduce the limit of the sum of (-1)^q/q and (-1)^q/(2q+1) You must be familiar with the technique of integration by parts and the antiderivatives of common functions to be able to do this exercise! ▬▬▬▬▬▬▬▬ KEYWORDS ▬▬▬▬▬▬▬▬ #Sequence_Defined_by_an_Integral #Integral_Calculation #Homework