Prove inequality about sequences

In summary, the statement is true and can be proven by induction, with the assumption that \sum_{n=1}^{\infty} v_n converges. By setting M = (1+\sum_{k=1}^{\infty} v_k), we can show that u_n \leq M for all natural number n.
  • #1
mercedesbenz
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Homework Statement


Let [tex]u_n, v_n[/tex] be a sequence of positive real number such that
[tex]u_{n+1}\leq (1+v_n)u_n[/tex] where [tex]\sum_{n=1}^{\infty}v_n[/tex]
converges.

It is true or false if true please help me prove.
if [tex]\frac{u_{n+1}}{u_n}\leq M [/tex] then [tex]u_n\leq M[/tex]
for some [tex]M[/tex] be a positive real number.

whether we can find [tex]M[/tex] is a positive real number such that
[tex]u_n\leq M[/tex] for all natural number n


Homework Equations





The Attempt at a Solution




 
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  • #2
Yes, it is true. We prove by induction. For n = 1, \frac {u_2}{u_1} \leq (1+v_1) \leq M for some positive real number M. Thus u_1 \leq M. Suppose that u_n \leq M for some n. Then \frac {u_{n+1}}{u_n} \leq (1+v_n) \leq (1+\sum_{k=1}^n v_k). Since \sum_{k=1}^{\infty} v_k converges, there exists a positive real number M such that (1+\sum_{k=1}^n v_k) \leq M. Thus u_{n+1} \leq Mu_n \leq M^2. Hence the result follows.
 

1. What is a sequence?

A sequence is a list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term, and the sequence can be finite (with a specific number of terms) or infinite (with an endless number of terms).

2. How do you prove an inequality about sequences?

To prove an inequality about sequences, you must show that the inequality holds for every term in the sequence. This can be done through mathematical induction, where you first show that the inequality holds for the first term, and then assume it holds for the nth term and use that to prove it holds for the (n+1)th term. This process is repeated until the inequality is proven for all terms in the sequence.

3. What is the importance of proving inequalities about sequences?

Proving inequalities about sequences is important in many areas of mathematics and science. It allows us to make accurate predictions and conclusions about the behavior of sequences, and can also be used to solve problems and make decisions in real-world situations.

4. Can inequalities about sequences only be proven for numerical sequences?

No, inequalities about sequences can also be proven for sequences of other mathematical objects, such as functions, matrices, or vectors. The same principles of mathematical induction and proving the inequality for each term in the sequence apply.

5. Are there any common techniques or strategies for proving inequalities about sequences?

Yes, there are several common techniques and strategies for proving inequalities about sequences, such as using algebraic manipulations, applying mathematical inequalities (such as the AM-GM inequality), or using calculus methods (such as using derivatives to show that the sequence is increasing or decreasing). It is important to choose the most appropriate technique based on the specific sequence and inequality being proven.

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