Limit of Series for Infinity Sum: Basic Comparison Test

[teng125]

for infinity sum n=1 (2n+1) / (5n+1)

how to find the converges or diverges??
is it suitable to use basic comparison test??pls show me

 

[TD]

If we split the fraction, we get

\frac{{2n + 1}}{{5n + 1}} = \frac{2}{5} + \frac{3}{{25n + 5}}

As you can see, the second term will go to 0 if n tends to infinity but there's a remaining term 2/5. In other words: the limit of the associated sequence isn't 0 (for n going to infinity) and this is a necessary (though not sufficient) condition for the convergence of the series.

 

[HallsofIvy]

Or: divide both numerator and denominator by n:
\frac{2n+1}{5n+1}= \frac{2+\frac{1}{n}}{5+\frac{1}{n}}

Now what does 1/n go to as n goes to infinity?

 

[teng125]

but the question ask for diverges or converges??and the answer is diverges

 

[TD]

When a series doesn't converge, it diverges.

 

[teng125]

is it necessary to use any rules such as the one i mentioned??or just to perform the steps showed above??

 

[TD]

I don't think those are necessary, if the limit of the (positive) sequence doesn't go to 0 (which is easy to check without using those other tests), then the (positive) series doesn't converge.

 

[Valhalla]

teng the n-th term test for divergence is an easy check for this one.