Can someone show me a step by step process and explanation how to solve this problem?
1) Consider the following series.
(∞ on top of summation symbol) (k = 1 under the summation symbol) ∑ k(k+15)/(k+13)^2
Determine whether the series is convergent or divergent. If it is convergent, find its sum.
To determine whether the series is convergent or divergent, we can use the limit comparison test.
First, let's simplify the expression inside the summation symbol: k(k+15)/(k+13)^2
We can rewrite this expression as k(k+15)/(k+13)(k+13):
Now, we can use the limit comparison test by comparing it to a known series. We choose the series ∑ 1/n^2, which is known to be convergent.
We can write the general term of the series we're considering as a[n] = k(k+15)/(k+13)(k+13), and the general term of the known series as b[n] = 1/n^2.
Taking the limit as n approaches infinity of the ratio a[n]/b[n], we get:
lim(n→∞) (k(k+15)/((k+13)(k+13))) / (1/n^2)
Let's simplify this limit:
lim(n→∞) (k(k+15)n^2) / ((k^2 + 26k + 169))
Now, we can cancel out the k and n^2 terms:
lim(n→∞) (k+15) / (k^2 + 26k + 169)
Now, let's take the limit as k approaches infinity:
lim(k→∞) (k+15) / (k^2 + 26k + 169)
To find the limit as k approaches infinity, we can divide all terms by k^2:
lim(k→∞) (1/k + 15/k^2) / (1 + 26/k + 169/k^2)
Now, taking the limit as k approaches infinity, we can ignore the terms that go to 0:
lim(k→∞) (0 + 0) / (1 + 0 + 0)
The limit is equal to 0.
According to the limit comparison test, if the limit is finite and positive, the two series converge or diverge together. If the limit is 0 or infinite, the series may or may not converge.
In this case, the limit is 0, so we cannot determine whether the series ∑ k(k+15)/(k+13)^2 converges or diverges using the limit comparison test. We need to use another test, such as the ratio test or the integral test, to make a conclusion about convergence.