(1+1/n)^-n^2 use root test
so, did you use the root test?
what did it tell you?
it told me to use the root test
I mean, did you actually apply the root test?
If so, what did you learn from it?
To determine the convergence or divergence of the series formed by the expression (1+1/n)^-n^2 using the root test, follow these steps:
Step 1: Take the nth root of the absolute value of the given expression.
Let's denote the given expression as a_n:
a_n = (1+1/n)^-n^2
Now, take the nth root on both sides of the equation:
(abs(a_n))^(1/n) = abs((1+1/n)^-n^2)^(1/n) [Taking nth root on both sides]
Step 2: Simplify the expression inside the absolute value and exponent.
Simplifying the exponent, we get:
(abs(a_n))^(1/n) = abs(1/n^2)^(-n)
Now, simplify the expression within the absolute value by raising it to the power of -n:
(abs(a_n))^(1/n) = abs(n^2)^(-n)
(abs(a_n))^(1/n) = abs(1/n^2n) [Simplifying n^2n]
Step 3: Evaluate the limit as n approaches infinity.
Taking the limit as n approaches infinity will help determine whether the series converges or diverges.
lim(n->∞) (abs(a_n))^(1/n) = lim(n->∞) (abs(1/n^2n))
To evaluate this limit, we can simplify the expression further:
lim(n->∞) (abs(a_n))^(1/n) = lim(n->∞) abs(1/n^2n)
lim(n->∞) (abs(a_n))^(1/n) = lim(n->∞) 1/n^2n
Now, taking the limit, we can see that the fraction 1/n^2n approaches 0 as n goes to infinity.
lim(n->∞) (abs(a_n))^(1/n) = 0
Step 4: Analyzing the limit result.
According to the root test, if the limit is less than 1, the series converges. If the limit is greater than 1 or infinity, the series diverges. If the limit is equal to 1, the test is inconclusive.
In this case, since the limit is 0, which is less than 1, we can conclude that the series formed by (1+1/n)^-n^2 converges.
Therefore, using the root test, we have determined that the series converges.