Series - Convergent or Divergent?

[CalculusGuy25]

Is the series convergent or divergent?

n=1 summation and it goes to infinity n!/2n!+1

[Infinite series]

Homework Equations

None.

The Attempt at a Solution

I have no idea.

 

[Mark44]

Is the series convergent or divergent?

n=1 summation and it goes to infinity n!/2n!+1

[Infinite series]

Homework Equations

None.

The Attempt at a Solution

I have no idea.

Before we can give you some help, you need to have made an effort at solving the problem you posted - forum rules. Since you are investigating the convergence/divergence of a series, some tests that can be used should have been presented in your textbook and in lecture. Have you tried any of those tests?

 

[CalculusGuy25]

Sorry.

I did some research and it said whenever there are factorials, use the ratio test. But, I haven't even learned that yet. I couldn't even find a problem similar to mine in the textbook. Even if you can't show me how to solve the problem, can you confirm the ratio test is the way to go?

 

[icystrike]

The 1 is in the denominator or beside? If it is beside , let us sum a infinite number of 1 , we will get a infinity big number won't we?

 

[Fredrik]

Is this what you meant?

$$\sum_{n=0}^\infty}\frac{n!}{2n!+1}$$

One way to proceed is to figure out what happens to the nth term as n→∞, or find an upper or lower bound to each term that simplifies the problem so much that you can immediately see if the series converges or not.

 

[HallsofIvy]

The factorials are a red herring. What happens to
$$\frac{a}{a+ 1}$$
as a goes to infinity?

 

[CalculusGuy25]

Is this what you meant?

$$\sum_{n=0}^\infty}\frac{n!}{2n!+1}$$

One way to proceed is to figure out what happens to the nth term as n→∞, or find an upper or lower bound to each term that simplifies the problem so much that you can immediately see if the series converges or not.

Yeah, that's what I meant. So, the ratio test would not be needed then?

The factorials are a red herring. What happens to
$$\frac{a}{a+ 1}$$
as a goes to infinity?

Oh, so you can simply ignore the factorials and examine what happens as you plug in numbers?

 

[Dickfore]

The factorials are a red herring. What happens to
$$\frac{a}{2a+ 1}$$
as a goes to infinity?

Fixed.

 

[Fredrik]

So, the ratio test would not be needed then?

In this case, it just makes the problem harder.

Oh, so you can simply ignore the factorials and examine what happens as you plug in numbers?

The point is that if you understand what happens to a/(a+1) as a goes to infinity, you should also understand what happens to 2n!/(2n!+1) as n goes to infinity. (The method that tells you what happens in the first case also tells you what happens in the second case).

 

[Dickfore]

The rule that allows you to do that is the continuity of the function:

$$f(a) = \frac{a}{2 a + 1}$$

and the sequence definition of a limit.

 

[Fredrik]

CalculusGuy25, have you figured this one out yet? I would like to show you my solution, but I can't until you've come up with one of your own. I can only give you hints until then. Is there anything you can tell us about the size of the nth term?

 

[gomunkul51]

http://en.wikipedia.org/wiki/Term_test

Read, Understand, Use.

 

[malicx]

Is the series convergent or divergent?

n=1 summation and it goes to infinity n!/2n!+1

[Infinite series]

Homework Equations

None.

The Attempt at a Solution

I have no idea.

I'd like to point out that (2n)! and 2(n!) are completely, completely different things...