Why does ∑(aₙ+4) diverge if ∑aₙ converges?

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

Suppose [itex]\sum_{n=1}^\infty a_n[/itex] converges. Determine the convergence of [itex]\sum_{n=1}^\infty a_n+4[/itex]

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.
 

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


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.
 


@micromass, the lim would be a finite number?
 


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


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


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

Depends on whether you mean

[tex]\sum (a_n + 4)[/tex]

or

[tex]\left(\sum a_n\right) +4[/tex]
 


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


IntegrateMe said:
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

[tex]\lim_{n\rightarrow +\infty}{a_n+4}[/tex]

??
 


I'd think it would be 4?
 


I guess I was right lol
 


IntegrateMe said:
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.
 


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