- #1
MillerGenuine
- 64
- 0
Homework Statement
[tex]
\sum_{n=1}^\infty \frac{1+4^n}{1+3^n}
where a_n= \frac{1+4^n}{1+3^n} and b_n= \frac{4^n}{3^n}
[/tex]
Homework Equations
I know how to do this problem, you take the limit as "n" goes to infinity of a_n/b_n ... which after a good amount of algebra ends up being 1. which is greater than zero. once i do this I go on to the next step which is...
[tex]
\sum_{n=1}^\infty \frac{4^n}{3^n}
[/tex]
ok so now once i get this I see that it is a geometric series where r= 4/3 which is greater than 1..which means the series Diverges by limit comparison test.
Easy enough probelm..but my question is what's with the hole limit comparison test portion? can't i just see my b_n= 4^n/3^n and therefore seeing its a geometric series, skipping the limit test all together and moving straight to Direct Comparison Test.. the second step i showed above. I tried doing this (skipping limit test) for a few problems in my book and they ended up being the same...so please tell me..whats with the foreplay of finding the limit, when it seems i can just go straight to Direct Comparison test by using my b_n to see if its a Geometric series or P-series?