# Short problem, but why is this the answer? Confused

1. Mar 12, 2012

### IntegrateMe

Short problem, but why is this the answer? Confused :(

Suppose $\sum_{n=1}^\infty a_n$ converges. Determine the convergence of $\sum_{n=1}^\infty a_n+4$

The answer is "divergent," but I don't see why that's necessarily true. I would assume we wouldn't know whether a_n + 4 is convergent/divergent.

2. Mar 12, 2012

### micromass

Staff Emeritus
Re: Short problem, but why is this the answer? Confused :(

If $\sum_{n=1}^{+\infty} a_n$ converges, then what is $\lim_{n\rightarrow +\infty}{ a_n}$??

3. Mar 12, 2012

### Mining_Engr

Re: Short problem, but why is this the answer? Confused :(

If the first series you have there converges, the second one also converges. Adding or subtracting a finite number from a convergent series cannot make it divergent.

4. Mar 12, 2012

### IntegrateMe

Re: Short problem, but why is this the answer? Confused :(

@micromass, the lim would be a finite number?

5. Mar 12, 2012

### Mining_Engr

Re: Short problem, but why is this the answer? Confused :(

Remember the test for divergence, if a series converges, the lim of the sequence must be zero at infinity.

6. Mar 12, 2012

### IntegrateMe

Re: Short problem, but why is this the answer? Confused :(

Ok, that makes sense. Does this prove that my answer is correct, or does the book answer still hold?

7. Mar 12, 2012

### micromass

Staff Emeritus
Re: Short problem, but why is this the answer? Confused :(

Depends on whether you mean

$$\sum (a_n + 4)$$

or

$$\left(\sum a_n\right) +4$$

8. Mar 12, 2012

### IntegrateMe

Re: Short problem, but why is this the answer? Confused :(

The first one, which I assume makes the book answer correct.

9. Mar 12, 2012

### micromass

Staff Emeritus
Re: Short problem, but why is this the answer? Confused :(

OK. For the first one, what is the limit of the term sequence?? What is

$$\lim_{n\rightarrow +\infty}{a_n+4}$$

??

10. Mar 12, 2012

### IntegrateMe

Re: Short problem, but why is this the answer? Confused :(

I'd think it would be 4?

11. Mar 12, 2012

### IntegrateMe

Re: Short problem, but why is this the answer? Confused :(

I guess I was right lol

12. Mar 12, 2012

### micromass

Staff Emeritus
Re: Short problem, but why is this the answer? Confused :(

Yes!! So can the series possibly converge?? Remember that if it converged then the limit would be 0.

13. Mar 12, 2012

### IntegrateMe

Re: Short problem, but why is this the answer? Confused :(

Ahh, ok, now I understand. Thank you for the help :D

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