Convergence of Series (Harder)

In summary, convergence of a series is the process of determining whether the infinite sum of terms in a series approaches a finite value as the number of terms approaches infinity. There are two types of convergence - absolute and conditional - and various methods for testing convergence, including the ratio test, root test, and comparison test. The difference between convergence and divergence is that convergence leads to a finite sum while divergence leads to infinity or oscillation. The convergence of a series is important in many applications, including physics, engineering, finance, and numerical approximations and calculations.
  • #1
sid9221
111
0
Prove that:
[tex] (1-\frac{1}{n})^n \rightarrow \frac{1}{e} [/tex] as [tex] n \to \infty [/tex]

you may use the fact that

[tex] (1+\frac{1}{n})^n \rightarrow e [/tex]

I have no idea where to even begin, can someone point me in the right direction ?
 
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  • #2
Try this: Multiply both sides by e.
[tex](1-\frac{1}{n})^n .e \rightarrow \frac{1}{e} .e[/tex]
[tex](1-\frac{1}{n})^n .(1+\frac{1}{n})^n \rightarrow 1[/tex]
If you can simplify the L.H.S. to get 1 on the R.H.S., then you have proved the convergence of the series.
 
  • #3
sharks said:
Try this: Multiply both sides by e.
[tex](1-\frac{1}{n})^n .e \rightarrow \frac{1}{e} .e[/tex]
[tex](1-\frac{1}{n})^n .(1+\frac{1}{n})^n \rightarrow 1[/tex]
If you can simplify the L.H.S. to get 1 on the R.H.S., then you have proved the convergence of the series.

Really don't know if you can sub in a sequence like that besides I need to prove it converges to [tex] \frac{1}{e}[/tex] not that it simply converges.
 
  • #4
[tex](1-\frac{1}{n})^n .(1+\frac{1}{n})^n \rightarrow 1[/tex]
[tex]( (1-\frac{1}{n}).(1+\frac{1}{n})) ^n \rightarrow 1[/tex]
[tex](1-\frac{1}{n}+\frac{1}{n}-\frac{1}{n^2})^n \rightarrow 1[/tex]
[tex](1-\frac{1}{n^2})^n \rightarrow 1[/tex]
[tex]\lim_{n\to \infty}(1-\frac{1}{n^2})^n=1[/tex]
 
  • #5
sid9221 said:
Prove that:
[tex] (1-\frac{1}{n})^n \rightarrow \frac{1}{e} [/tex] as [tex] n \to \infty [/tex]

you may use the fact that

[tex] (1+\frac{1}{n})^n \rightarrow e [/tex]

I have no idea where to even begin, can someone point me in the right direction ?
Let ##a_n = \left(1 - \frac{1}{n}\right)^n##. Try determining what ##\log a_n## converges to.
 

FAQ: Convergence of Series (Harder)

Question 1: What is the definition of convergence of a series?

The convergence of a series is the process of determining whether the infinite sum of terms in a series approaches a finite value as the number of terms approaches infinity. In other words, it is a way to determine if the sum of a series will eventually reach a constant value or if it diverges to infinity.

Question 2: What is the difference between absolute and conditional convergence?

Absolute convergence occurs when the series converges regardless of the order in which the terms are added. Conditional convergence, on the other hand, only occurs when the series converges if the terms are added in a specific order. In other words, the rearrangement of terms can change the value of a conditionally convergent series, but not an absolutely convergent one.

Question 3: How do you test for convergence of a series?

There are various methods for testing convergence of a series, including the ratio test, the root test, and the comparison test. These tests involve comparing the given series to a known convergent or divergent series or evaluating the limit of a specific ratio or root. Ultimately, the goal is to determine if the series approaches a finite value, diverges to infinity, or oscillates between different values.

Question 4: What is the difference between convergence and divergence?

Convergence refers to the process of approaching a finite value as the number of terms in a series increases, while divergence refers to the process of approaching infinity or oscillating between different values as the number of terms increases. In other words, a series that converges has a finite sum, while a series that diverges does not.

Question 5: Why is the convergence of a series important?

The convergence of a series is important because it allows us to determine the behavior of an infinite sum of terms. This is useful in many applications, such as physics, engineering, and finance, where infinite series are used to model real-world phenomena. Additionally, the convergence of a series is essential in determining the accuracy and reliability of numerical approximations and calculations.

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