Short problem, but why is this the answer? Confused

[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.

 

[micromass]

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

 

[Mining_Engr]

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.

 

[IntegrateMe]

@micromass, the lim would be a finite number?

 

[Mining_Engr]

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

 

[IntegrateMe]

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

 

[micromass]

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

Depends on whether you mean

$$\sum (a_n + 4)$$

or

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

 

[IntegrateMe]

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

 

[micromass]

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

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

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

??

 

[IntegrateMe]

I'd think it would be 4?

 

[IntegrateMe]

I guess I was right lol

 

[micromass]

I'd think it would be 4?

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

 

[IntegrateMe]

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