Determine whether the series is convergent or divergent.∑from k=1to ∞ (ke^-9k)
since e^(-9k) goes to zero much faster than k grows, the series converges.
Use the ratio test.
A_(n+1)/A_n = (k+1)e^(-9k-9)/ke^(-9k)
= (k+1)/k * e^-9
To determine whether the series ∑(k=1 to ∞) (k*e^(-9k)) is convergent or divergent, we can use the ratio test.
The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series is convergent. On the other hand, if the limit is greater than 1 or undefined, the series is divergent.
Let's apply the ratio test to our series:
∑(k=1 to ∞) (k*e^(-9k))
First, let's consider the ratio of consecutive terms:
R = [(k+1)*e^(-9(k+1))] / [k*e^(-9k)]
Next, let's simplify the expression:
R = [(k+1)/k] * [e^(-9(k+1))/e^(-9k)]
R = (k+1)/k * e^(-9k-9)
Now, let's take the limit of the absolute value of the ratio as k approaches infinity:
lim[k->∞] |R| = lim[k->∞] |(k+1)/k * e^(-9k-9)|
Applying the limit:
lim[k->∞] |R| = lim[k->∞] (k+1)/k * e^(-9k-9)
lim[k->∞] |R| = lim[k->∞] (1 + 1/k) * e^(-9k-9)
As k approaches infinity, the term (1/k) approaches 0. The term e^(-9k-9) also approaches 0 as k approaches infinity. Therefore, the limit of |R| as k approaches infinity is 0.
Since the limit of |R| is less than 1, we can conclude that the series ∑(k=1 to ∞) (k*e^(-9k)) is convergent.
To determine if the given series is convergent or divergent, we can use the ratio test.
The ratio test states that for a series ∑a_k, if the limit of the absolute value of the ratio of consecutive terms (as k approaches infinity) is less than 1, then the series is convergent. If the limit is greater than 1 or does not exist, then the series is divergent.
Let's apply the ratio test to the given series:
a_k = k * e^(-9k)
We need to calculate the limit of the absolute value of the ratio of consecutive terms:
Lim (k→∞) |(a_(k+1))/(a_k)|
Taking the ratio of consecutive terms:
|(a_(k+1))/(a_k)| = |((k+1) * e^(-9(k+1)))/(k * e^(-9k))|
Simplifying the expression, we get:
|(k+1) * e^(-9(k+1)))/(k * e^(-9k)) = |(k+1)/(k) * e^(-9(k+1)+9k)| = |(k+1)/(k) * e^(-9)|
Since e^(-9) is a constant, we can ignore it in the limit calculation and focus on |(k+1)/(k)|:
Lim (k→∞) |(k+1)/(k)|
The limit as k approaches infinity of (k+1)/(k) is 1. Therefore, the ratio of consecutive terms is 1.
Since the ratio is equal to 1, the limit is not less than 1. As per the ratio test, this implies that the given series is inconclusive.
In other words, the ratio test does not provide enough information to determine if the series is convergent or divergent. Additional tests or methods need to be applied to determine the convergence or divergence of the series.