I Does the sequence converge or diverge? (2^n)/(2n)

[Salman Ali]

TL;DR Summary

Which method easier to solve such questions which involve factorials?

So there are two parts of the question:
a) does the sequence converge or diverge
b) use nth term on the Series
Now sybomlab calculator is saying to apply ratio test!
a)

b)
So should I apply ratio test or is there any easy method? And what's the difference between these two questions and what methods differ?

 

[PeroK]

Summary: Which method easier to solve such questions which involve factorials?

So there are two parts of the question:
a) does the sequence converge or diverge
b) use nth term on the Series
Now sybomlab calculator is saying to apply ratio test!
a)View attachment 249006

b)
So should I apply ratio test or is there any easy method? And what's the difference between these two questions and what methods differ?

What could be easier than the ratio test?

 

[jedishrfu]

You could factor out the 2 from the 2n and then the answer becomes obvious using the ratio test.

Be careful here to only factor out the 2’s that you need.

 

[Mark44]

And what's the difference between these two questions and what methods differ?

The first question asks whether the given sequence converges. That is, whether $\{\frac 1 {0!}, \frac 2 {2!}, \frac 4 {4!}, \dots \}$ converges.

The second question asks whether the series (the sum of the terms of the sequence above) converges. In other words, whether $\{\frac 1 {0!} + \frac 2 {2!} + \frac 4 {4!} + \dots \}$ converges. At this point in your studies it's important to understand the difference between these two words.

 

[Salman Ali]

The first question asks whether the given sequence converges. That is, whether $\{\frac 1 {0!}, \frac 2 {2!}, \frac 4 {4!}, \dots \}$ converges.

The second question asks whether the series (the sum of the terms of the sequence above) converges. In other words, whether $\{\frac 1 {0!} + \frac 2 {2!} + \frac 4 {4!} + \dots \}$ converges. At this point in your studies it's important to understand the difference between these two words.

Can you kindly explain how to solve both of them ?

 

[PeroK]

Can you kindly explain how to solve both of them ?

We can't do that. You need to post your best effort.

 

[Mark44]

Can you kindly explain how to solve both of them ?

The ratio test, which you already mentioned, is easy to use to determine whether the sequence converges.
Your textbook should have several tests you can use to determine whether a series converges, as well as examples of how to use them.

b) use nth term on the Series

The nth term test is often not useful. It can be used to determine that a series diverges, but it doesn't tell you when a series converges.
The full name of the test is "nth term test for divergence."

 

[Salman Ali]

We can't do that. You need to post your best effort.

 

[Salman Ali]

The ratio test, which you already mentioned, is easy to use to determine whether the sequence converges.
Your textbook should have several tests you can use to determine whether a series converges, as well as examples of how to use them.

The nth term test is often not useful. It can be used to determine that a series diverges, but it doesn't tell you when a series converges.
The full name of the test is "nth term test for divergence."

I am bound to use nth term for the second part. Its mentioned in the question. I'll give it a try.

 

[PeroK]

The ratio test determines whether a series converges.

What can you say about the sequence?