Is the series 1/2n convergent or divergent?

[srfriggen]

Q: Is the series 1/2n convergent or divergent? If convergent find it's sum.

I see that the limit as n goes to infinity is zero but that does not prove C or D. It seems like it would be divergent but why? How can that be shown?

Any direction/help would be greatly appreciated.

 

[Dick]

Try an integral test. Do you know that one?

 

[srfriggen]

Yes I do but I shouldn't lol (we haven't gotten to that in class). We have covered the harmonic series and shown it is D.

I also notice I can't do a comparison to 1/n since 1/n > 1/2n.
Would the argument 1/2 * 1/n = 1/2 x divergent = D make sense?

 

[Dick]

Yes I do but I shouldn't lol (we haven't gotten to that in class). We have covered the harmonic series and shown it is D.

I also notice I can't do a comparison to 1/n since 1/n > 1/2n.
Would the argument 1/2 * 1/n = 1/2 x divergent = D make sense?

Sure. If a series is divergent, then any nonzero constant times the series is also divergent. That's true. But do you have a theorem or something to justify it rather than just writing divergent*(1/2)=divergent? That just seems a little sloppy.

 

[srfriggen]

Yeah I know. I'm typing on this forum from my phone so it's tough to be elegant lol. The question only asks C or D but I kinda wanted to know for my own purposes but thanks for the help in understanding it. :)