use the root test to to determine the convergence or divergence of the series 100n/e^n
so, what about the root test is causing difficulty?
I am having problem with the 1oo n. I know by taking the limit to the 1/n power gives me e in the denominator. How and what does (100n)^1/2n become?
To determine the convergence or divergence of the series 100n/e^n using the Root Test, follow these steps:
Step 1: Write down the general term of the series. In this case, the general term is given as 100n/e^n.
Step 2: Apply the Root Test formula. The Root Test states that if the limit of the nth root of the absolute value of the general term is less than 1, then the series converges. If the limit is greater than 1 or undefined, then the series diverges.
Step 3: Find the limit of the nth root of the absolute value of the general term as n approaches infinity.
Let's calculate the limit using the Root Test:
lim┬(n→∞)〖(100n/e^n )^(1/n) 〗
Step 4: Simplify the expression inside the limit:
(100n/e^n )^(1/n) = 100^(1/n) * n^(1/n) / e
Step 5: Evaluate the limit as n approaches infinity:
lim┬(n→∞)〖(100n/e^n )^(1/n) 〗 = lim┬(n→∞)(100^(1/n) * n^(1/n) / e) = 1 * 1 / e = 1/e
Step 6: Determine the convergence or divergence based on the limit:
Since the limit of (100n/e^n)^(1/n) is less than 1, specifically 1/e, the series 100n/e^n converges.
Therefore, the series 100n/e^n converges.