Determine whether the series converges or diverges.
[tex]\sum[/tex](1+5^n) / (1+6^n)
limit comparison test, or just the comparison test.
The Attempt at a Solution
I have a couple of ways of trying to proves this.
1) (1+5^n) / (1+6^n) < (1 + 5^n) / 6^n =
(1/6)^n + (5/6)^n , which converges to 1/5 + 5 = 26/5
so since the series is bounded by 0 <= (1+5^n)/(1+6^n) <=26/5
it converges? This might not necessarily be true because a function might fluctuate between
those points until infinity, but I think that only cosine have this property. Thus this series converges.
[attempt # 2]
(1+5^n) / (1+6^n) =
1 / (1+6^n) + (5/6)^n
I know that (5/6)^n converges to 5, so
= 1/(1+6^n) + 5
now I apply the comparison test to 1/(1+6^n)
1/(1+6^n) < 1/(6^n), and this converges to 0.2
[0<= 1/(1+6^n) <= 0.2 ] + 5
so it converges since 1/(1+6^n) is bounded.
Either 1 right?