∞
Σ n/(√n^7+5n+1)
n=1
I just tried using a sub n and b sub n and got the limit = 1, therefore it diverges right? but I'm not sure
To determine whether the series ∑(n/(√n^7 + 5n + 1)) as n approaches infinity converges or diverges, you can use the Ratio Test.
Let's start by finding the ratio of consecutive terms:
|((n+1)/√(n+1)^7 + 5(n+1) + 1)| / |(n/√n^7 + 5n + 1)|
Simplifying this expression will help us determine its behavior at n approaches infinity.
First, let's simplify the expression inside the absolute values by substituting n with n+1:
√((n+1)^7 + 5(n+1) + 1) / √n^7 + 5n + 1
Next, multiply the numerator and denominator by √n to eliminate the square root:
√((n+1)^7 + 5(n+1) + 1) * √n / √n^7 + 5n + 1 * √n
Simplifying further, we can apply the product rule for radicals:
√((n+1)^7 * n) + 5(n+1)√n + √n / (√n * √n^7) + 5n√n + √n
Now, let's distribute the radical and simplify:
√(n^(7+1) + 7n^(7) + 21n^(6) + 35n^(5) + 30n^(4) + 15n^(3) + 5n^(2) + n + n + 5) / (n^(7/2) + 5n^(3/2) + n^(1/2))
Simplifying the formula:
√(n^8 + 7n^7 + 21n^6 + 35n^5 + 30n^4 + 16n^2 + 6n + 5) / n^(7/2) + 5n^(3/2) + n^(1/2)
To analyze this expression as n approaches infinity, we need to determine the exponent with the highest power of n in both the numerator and denominator:
In the numerator, the highest power of n is n^8, while in the denominator, the highest power is n^(7/2). Since the exponent in the denominator is greater, we can ignore all other terms in the numerator and rewrite the expression:
(√n^8) / √n^(7/2)
Simplifying:
n^4 / n^(7/4) = n^(4 - 7/4) = n^(16/4 - 7/4) = n^(9/4)
Now, let's analyze the behavior of n^(9/4) as n approaches infinity.
If the exponent is greater than 1, n^(9/4) would tend to infinity as n approaches infinity, indicating the series would diverge.
However, since the exponent (9/4) is less than 1, n^(9/4) would approach 0 as n approaches infinity. This suggests that the terms of the series ∑(n/(√n^7 + 5n + 1)) as n approaches infinity would tend to 0.
Since the terms of the series approach 0, the series likely converges. To confirm this, we need to perform additional steps.
You mentioned that when you tried a substition using sub n and b sub n, you got the limit = 1. However, a limit of 1 does not indicate that the series necessarily diverges. The limit of the terms should approach 0 for convergence.
Therefore, based on the Ratio Test and the analysis of the terms, it seems like the series ∑(n/(√n^7 + 5n + 1)) as n approaches infinity converges.