Math Problem Statement

Solve the given set of exercises from a mathematics and computer science sheet, which includes exercises on sequence calculations, limits, recursive sequences, and convergence.

Solution

Let's go through each exercise one by one and tackle the problems.

Exercise 1

Calculate U1U_1, U2U_2, and U3U_3 for the following cases:

  1. Un=n+1nsinnπ2U_n = \frac{n+1}{n} \sin \frac{n \pi}{2}
  2. Un=5n×9(5+2n)4×7×10(4+3n)U_n = \frac{5n \times 9 \cdots (5+2n)}{4 \times 7 \times 10 \cdots (4+3n)}
  3. Un=1nk=1nk2U_n = \frac{1}{n} \sum_{k=1}^n k^2

For each part, we will substitute values for n=1n = 1, n=2n = 2, and n=3n = 3 to compute U1U_1, U2U_2, and U3U_3.

Exercise 2

Calculate the following limits:

  1. limn(1+12+13++1n(n+1))\lim_{n \to \infty} \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n(n+1)}\right)
  2. limnn2+7n4+n+1\lim_{n \to \infty} \frac{n^2 + 7}{\sqrt{n^4 + n + 1}}
  3. limnn2+9n2+4\lim_{n \to \infty} \sqrt{n^2 + 9} - \sqrt{n^2 + 4}

These problems typically require simplifying the expressions inside the limits or using known limit properties and theorems.

Exercise 3

The sequence (Un)(U_n) is defined by Un=ln(1+Un1)U_n = \ln(1 + U_{n-1}) with U0>0U_0 > 0. Find the limit of (Un)(U_n).

This exercise involves finding the behavior of the recursive sequence UnU_n as nn approaches infinity, likely requiring us to assume the existence of a limit and solving it based on the recursion.

Exercise 4

  1. Let (Un)(U_n) be an increasing sequence. Show that the sequence (Vn)=U1+U2++Unn(V_n) = \frac{U_1 + U_2 + \cdots + U_n}{n} is also increasing.
  2. If (Un)(U_n) is a convergent sequence, can we deduce that (Vn)(V_n) is also convergent?

These parts involve proving monotonicity and convergence properties for the sequences (Un)(U_n) and (Vn)(V_n).

Exercise 5

Given U1=45U_1 = \frac{4}{5}, U2=134221U_2 = \frac{1 \cdot 3}{4 \cdot 2^2 1}, and for all n3n \geq 3, Un=13(2n1)4nn!U_n = \frac{1 \cdot 3 \cdots (2n - 1)}{4^n n!}:

  1. Show the inequality Un+1Un<12\frac{U_{n+1}}{U_n} < \frac{1}{2}.
  2. Deduce the limit of the sequence (Un)(U_n).

This requires calculating the ratio Un+1Un\frac{U_{n+1}}{U_n} and investigating the behavior of UnU_n as nn \to \infty.

Exercise 6

  1. Let qq be a real number such that q<1|q| < 1. Show that limnqn=0\lim_{n \to \infty} q^n = 0.
  2. Let Sn=q+q2+q3++qnS_n = q + q^2 + q^3 + \cdots + q^n. Calculate (1q)Sn(1 - q) S_n and then find limnSn\lim_{n \to \infty} S_n.

This is a classic problem involving geometric series and their sums as nn approaches infinity.

Exercise 7

Consider the sequence defined by:

U_0 = 2 \\ U_{n+1} = \frac{1}{2} U_n + \frac{1}{U_n}, \quad n \in \mathbb{N}. \end{cases}$$ This problem involves analyzing the recursive sequence $$U_n$$ and possibly finding its behavior or limit. --- Would you like a step-by-step solution for a specific exercise, or do you have a particular part you want to discuss in detail? --- ### Suggested Follow-Up Questions 1. What techniques are commonly used for calculating limits involving square roots, like in Exercise 2? 2. How can recursion formulas, like in Exercise 3, reveal the limit of a sequence? 3. What approach is helpful to show that a sequence is increasing, as in Exercise 4? 4. How does proving an inequality, as in Exercise 5, help in understanding the convergence of a sequence? 5. Why does the geometric series limit in Exercise 6 rely on the condition $$|q| < 1$$? ### Tip For sequences defined recursively, assume the sequence converges to a limit and substitute it back into the recursion to solve for possible limit values. This often simplifies the problem.

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Math Problem Analysis

Mathematical Concepts

Sequences and Series
Limits
Convergence
Recurrence Relations
Inequalities

Formulas

Sum of sequence terms
Limit of sequences
Recursive sequence definition
Geometric series sum formula
Inequality for sequence convergence

Theorems

Limit properties for sequences
Squeeze theorem
Monotonic sequence theorem
Geometric series convergence criterion

Suitable Grade Level

Undergraduate level (Mathematics/Computer Science)