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

#### Salman Ali

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?

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#### PeroK

Homework Helper
Gold Member
2018 Award
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

Mentor
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

Mentor
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

Homework Helper
Gold Member
2018 Award
Can you kindly explain how to solve both of them ?
We can't do that. You need to post your best effort.

#### Mark44

Mentor
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."

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#### 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.