Which of the following statements is true for the series the summation from n equals 0 to infinity of the nth power of the quotient of pi and e?
A. Converges to the quotient of e and the quantity e minus pi
B. Converges to the quotient of the quantity e minus pi and e
C. Converges to the quotient of the quantity e minus pi and pi
D. Diverges
To determine which statement is true for the given series, we can start by analyzing the convergence of the series.
The series is given by the sum from n equals 0 to infinity of (pi/e)^n.
This is a geometric series with the common ratio r = pi/e. A geometric series converges if and only if the absolute value of the common ratio is less than 1.
|pi/e| = pi/e is less than 1 since both pi and e are positive. Therefore, the series converges.
Now let's evaluate the convergence limit of the series.
The limit as n approaches infinity of (pi/e)^n is defined as follows:
lim (n → ∞) (pi/e)^n = 0, as |pi/e| < 1.
Since the limit of the series is 0, it means that the sum of the series converges to a finite value.
We can conclude that the series converges to a specific value.
Now let's go through the statements to determine which one is true:
A. Converges to the quotient of e and the quantity e minus pi
B. Converges to the quotient of the quantity e minus pi and e
C. Converges to the quotient of the quantity e minus pi and pi
D. Diverges
To find the value to which the series converges, we need to evaluate the sum.
The sum of a convergent geometric series can be calculated using the formula:
S = a / (1 - r), where a is the first term and r is the common ratio.
In our case, since a = 1 (the first term), and r = pi/e (the common ratio), we can substitute these values into the formula:
S = 1 / (1 - pi/e)
Now let's compare this result to the statements:
A. Converges to the quotient of e and the quantity e minus pi: This statement is not correct since the sum is in terms of pi and e, not e minus pi.
B. Converges to the quotient of the quantity e minus pi and e: This statement is not correct since the sum is in terms of pi and e, not e minus pi.
C. Converges to the quotient of the quantity e minus pi and pi: This statement is not correct since the sum is in terms of pi and e, not e minus pi.
D. Diverges: This statement is not correct since we have already determined that the series converges.
Therefore, none of the statements provided are true. The correct answer choice is not among the given options.