Determine the converge or divergence of the sequence

In summary, to determine the convergence of the sequence a_n = (1 + k/n)^n, the professor suggested converting it to f(x) and using ln (ln y) and L'Hospital's Rule. However, there is also another way to find the convergence by showing that the sequence b_n = (1 + 1/n)^n is bounded and increasing, and then using a subsequence of a_n to prove that the whole sequence converges. This approach may be more complicated than the professor's suggestion.
  • #1
merced
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[tex]a_n[/tex] = [tex](1 + k/n)^n[/tex]

Determine the converge or divergence of the sequence. If it is convergent, find its limit.

My professor said to convert the sequence to f(x) and use ln (ln y) and L'Hospital's Rule.

Do I have to use ln? Is there another way to find the convergence?
 
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  • #2
[tex] e \equiv \lim_{x\rightarrow \infty}(1+\frac{1}{x})^{x} [/tex]
 
  • #3
Oh, I didn't see that.
Thanks
 
  • #4
You could show that [itex]b_n=(1 + 1/n)^n[/itex] converges by showing it is bounded and increasing. This implies that for any k, [itex](b_n)^k[/itex] converges, right? Then, with [itex]c_n=kn[/itex], notice that your a_n is such that

[tex]a_{c_n}=(b_{n})^k[/tex]

That is to say, there is a subsequence of a_n that converges. This shows that a_n converges (towards the limit of [itex](b_n)^k[/itex]) because a_n is increasing, so it suffices to show that some subsequence converges to prove that the whole sequence does too.

But this is all probably more complicated then you prof's advice, eh?
 
Last edited:
  • #5
Yep...it worked the way my professor suggested.
 

1. What is a sequence?

A sequence is a list of numbers or terms that follow a specific pattern or rule. Each number in the sequence is called a term, and the position of the term in the sequence is called its index.

2. What is convergence and divergence?

In mathematics, convergence refers to a sequence of numbers that approaches a specific value or limit as the index increases. Divergence, on the other hand, means that the sequence does not have a specific limit and the terms either increase or decrease without bound.

3. How do you determine the convergence or divergence of a sequence?

To determine the convergence or divergence of a sequence, you can look at the behavior of the terms as the index increases. If the terms approach a specific value, the sequence is convergent. If the terms do not approach a specific value, the sequence is divergent.

4. What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series or sequence where the terms are always positive or always negative. Conditional convergence, on the other hand, refers to a series or sequence where the terms alternate between positive and negative. In conditional convergence, rearranging the terms can change the value of the sum, while in absolute convergence, rearranging the terms will not change the value of the sum.

5. What are some common tests used to determine convergence or divergence?

Some common tests used to determine convergence or divergence of a sequence include the ratio test, the root test, the comparison test, and the integral test. These tests compare the given sequence to a known convergent or divergent sequence to determine its behavior.

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