I need to find if the summation of (n^4)/(n^10 + 1) is convergent or divergent from n=1 to infinity.
I tried splitting it up into two sums, one being 1/n^6, which would be convergent because p=6>1, and then the other being n^4, but I'm not sure how to know if this is convergent or divergent.
I know the answer is convergent for the entire series because I first tried to answer divergent, but I don't know why it is convergent. Thanks!
it is clearly convergent, because
(n^4)/(n^10 + 1) < n^4/n^10 = 1/n^6
and 1/n^p is convergent for p>1
To determine if the given series is convergent or divergent, we can apply the Limit Comparison Test.
Step 1: Split the series into two parts.
Let's consider the series a_n = (n^4)/(n^10 + 1). We can split it into two parts as follows:
First part (1/n^6): We have correctly identified that the series ∑(1/n^6) is convergent because the exponent 6 is greater than 1 (p = 6 > 1).
Second part (n^4): We need to determine whether the series ∑(n^4) is convergent or divergent.
Step 2: Apply the Limit Comparison Test.
The Limit Comparison Test states that if we have two positive series a_n and b_n, and the limit of (a_n / b_n) as n approaches infinity equals a positive finite number L, then both series either converge or diverge.
Let's compare the series ∑(n^4) with the convergent series ∑(1/n^6) using the Limit Comparison Test.
Consider the limit as n approaches infinity of (a_n / b_n):
lim (n^4 / (1/n^6)) as n approaches infinity
= lim (n^4 * (n^6 / 1)) as n approaches infinity
= lim (n^10) as n approaches infinity.
As n approaches infinity, n^10 goes to infinity as well.
Therefore, the limit of (a_n / b_n) as n approaches infinity equals infinity (L = ∞). Since the limit gives a positive infinite value, the series ∑(n^4) has the same behavior as the series ∑(1/n^6), which we know is convergent.
Hence, the given series ∑(n^4)/(n^10 + 1) is also convergent.
Please note that this explanation assumes you have an understanding of series convergence tests and their properties.