Use the ratio test to find whether the series diverges or converges.
1/5^n (1 to infinity)
I think the limit converges to 1/5, so the series converges.
No, that is a different Anonymous. I'm still not sure about my answer.
an+1 = 1/5^(n+1)
an = 1/5^n
an+1/an = 5^n/5^(n+1) = 1/5 indeed
so it converges
thank you
To determine whether the series converges or diverges using the ratio test, we need to evaluate the limit of the ratio of consecutive terms as n approaches infinity.
Let's consider the series 1/5^n (1 to infinity). To apply the ratio test, we need to take the ratio of the (n+1)th term to the nth term:
a_n+1 / a_n = (1/5^(n+1)) / (1/5^n)
= 5^n / 5^(n+1)
= 5^n / (5^n * 5^1)
= 1 / 5
Taking the limit of this expression as n approaches infinity:
lim (n->infinity) a_n+1 / a_n = lim (n->infinity) 1 / 5 = 1/5
Since the limit (1/5) is less than 1, the series converges by the ratio test. Therefore, you are correct that the series converges.