Proof that a recursive sequence converges

In summary, Homework Statement Prove that ##\displaystyle t_{n+1} = (1 - \frac{1}{4n^2}) t_n## where ##t_1=1## converges.Homework EquationsThe Attempt at a SolutionIn summary, we have shown that the sequence is bounded below by 0 and is decreasing, thus it must converge by the Monotone Convergence Theorem.
  • #1
Mr Davis 97
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Homework Statement


Prove that ##\displaystyle t_{n+1} = (1 - \frac{1}{4n^2}) t_n## where ##t_1=1## converges.

Homework Equations

The Attempt at a Solution


First, we must prove that the sequence is bounded below. We will prove that it is bounded below by 0. ##t_1 = 1 \ge 0##, so the base case holds. Now, suppose that ##t_k \ge 0##. Then ##t_{k+1} = (1 - \frac{1}{4k^2}) t_k \ge (1 - \frac{1}{4k^2}) (0) = 0##. So the induction is complete, and the sequence is bounded below by 0.

Now we must show that the sequence is decreasing. ##t_2 = 15/16 \le 1 = t_1##, so the base case holds. Now, suppose that ##t_{k+1} \le t_k##. Then ##t_{k+2} = (1 - \frac{1}{4(k+1)^2})t_{k+1} \le t_{k+1}##. So the induction is complete, and the sequence is decreasing.

By the Monotone Convergence Theorem, the sequence converges.
 
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  • #2
Seems ok, btw you can actually solve this recursive equation by iteration, and see that it's decreasing explicitly.
 
  • #3
Mr Davis 97 said:

Homework Statement


Prove that ##\displaystyle t_{n+1} = (1 - \frac{1}{4n^2}) t_n## where ##t_1=1## converges.

Homework Equations

The Attempt at a Solution


First, we must prove that the sequence is bounded below. We will prove that it is bounded below by 0. ##t_1 = 1 \ge 0##, so the base case holds. Now, suppose that ##t_k \ge 0##. Then ##t_{k+1} = (1 - \frac{1}{4k^2}) t_k \ge (1 - \frac{1}{4k^2}) (0) = 0##. So the induction is complete, and the sequence is bounded below by 0.

Now we must show that the sequence is decreasing. ##t_2 = 15/16 \le 1 = t_1##, so the base case holds. Now, suppose that ##t_{k+1} \le t_k##. Then ##t_{k+2} = (1 - \frac{1}{4(k+1)^2})t_{k+1} \le t_{k+1}##. So the induction is complete, and the sequence is decreasing.

By the Monotone Convergence Theorem, the sequence converges.

Another, perhaps easier way is to note that
$$|t_{n+1}-t_n| = \frac{1}{4n^2} |t_n| \leq \frac{1}{4} |t_n|,$$
so we have (at least) geometrically fast convergence.
 
  • #4
One question. In showing that the sequence decreases, why didn't I have to use the inductive hypothesis?
 
  • #5
Mr Davis 97 said:
One question. In showing that the sequence decreases, why didn't I have to use the inductive hypothesis?
$$ \frac{t_{n+1}}{t_n }= 1 - \frac{1}{4n^2} ,$$
so if you know that ##t_n > 0## then you have ##0 < t_{n+1}/t_n < 1,## hence a positive strictly decreasing sequence.
 
  • #6
For [itex]n \geq 1[/itex], it follows from [tex]
\frac34 \leq 1 - \frac{1}{4n^2} < 1[/tex] that [tex]
0 < |t_{n+1}| = \left|1 - \frac{1}{4n^2}\right||t_n| < |t_n|[/tex] and so [itex]|t_n|[/itex] is strictly decreasing. Since each [itex]t_n[/itex] is by construction a product of strictly positive reals we can immediately remove the absolute value signs.
 

Related to Proof that a recursive sequence converges

1. What is a recursive sequence?

A recursive sequence is a sequence of numbers where each term is defined by a previous term. This means that the formula for each term in the sequence depends on the previous term(s).

2. How do you prove that a recursive sequence converges?

To prove that a recursive sequence converges, you can use the Monotone Convergence Theorem. This theorem states that if a sequence is both bounded and increasing (or decreasing), then it must converge. This means that if you can show that a recursive sequence is both bounded and increasing (or decreasing), then it must converge.

3. What does it mean for a recursive sequence to converge?

When a recursive sequence converges, it means that as the number of terms in the sequence increases, the terms get closer and closer to a specific value. This value is called the limit of the sequence.

4. How do you determine the limit of a convergent recursive sequence?

To determine the limit of a convergent recursive sequence, you can use the recursive formula to find the limit. This means that you substitute the limit into the recursive formula and solve for the limit. Alternatively, you can also use algebraic techniques, such as the Algebraic Limit Theorem, to find the limit.

5. Can a recursive sequence diverge?

Yes, a recursive sequence can also diverge. This means that as the number of terms in the sequence increases, the terms do not approach a specific value. Instead, they can either increase without bound (diverge to infinity) or fluctuate without a clear pattern (oscillate). In order to prove that a recursive sequence diverges, you can use the Divergence Test, which states that if the limit of the sequence does not equal 0, then the sequence must diverge.

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