Series question/comparison test

[Null_]

Homework Statement

The original series is the one on the left.

Homework Equations

The Attempt at a Solution

In the picture. I used the comparison test, and thought that since my original series, an, is less than the decreasing convergent series bn, that an is also convergent. There are a few problems like this that I seem to always mess up. It is divergent, but I would like an explanation.

Thanks.

edit- sorry for the large picture.

 

[blinktx411]

Why do you think the original series is divergent? Your analysis looks correct.

 

[flyingpig]

I recommend you try to break up the fraction like this

\sum_{n=1}^{\infty }\frac{ n -1}{n4^n} = \sum_{n=1}^{\infty }\frac{n}{n4^n} - \frac{1}{n4^n}

Now it is up to you to carry on

 

[Null_]

Why do you think the original series is divergent? Your analysis looks correct.

The answer key my professor gave us says it is divergent. There have been mistakes on them before, though..


Flyingpig, I tried your suggestion but am unable to get an answer. Other than taking the limit as n->infinity, which gives 0 and means nothing, I don't know what to do.

 

[flyingpig]

I must be an idiot to be doing math now because I got a chem exam in 30 mins lol

I'll give you another hint

Look at only

\sum_{n=1}^{\infty } \frac{-1}{n4^n}

Also, if the limit goes to 0, you can't conclude a thing about it...so don't fall into the pitfall and say "it is convergent!"

 

[micromass]

I suspect that your professor made a mistake. Your analysis looks correct...

 

[Null_]

I hope your chem test went well, flyingpig!

Thanks for the conformation, micromass. I was driving myself crazy trying to figure out what was wrong...

 

[flyingpig]

Sorry I take it back, I was rushing to answer when I was typing my answer. I thought it was n^n, but it was 4^n

Sorry lol