Which of the following is true for the series of summation from n=0 to infinity of(-1/x)^n
a) For x=2, the series diverges
b) For x=-1/2 , the series converges
c) For x=1/2, the series converges
d) For x=3, the series converges
Sub in the values for x then change the n values from (0, 1, 2...) and see what happens to the answers : )
I got answer A with your help. Is that correct?
To determine whether a series converges or diverges, we can make use of the ratio test. The ratio test states that for a series of the form ∑(aₙ), if the limit of the absolute value of the ratio of consecutive terms satisfies the condition:
lim(n→∞) |aₙ₊₁ / aₙ| < 1
then the series converges. If the limit is greater than 1 or does not exist, the series diverges.
Let's apply the ratio test to the given series:
∑((-1/x)^n), where n goes from 0 to infinity.
We can rewrite this series as:
∑((-1)^n / x^n).
Using the ratio test, let's find the limit of the absolute value of the ratio of consecutive terms:
lim(n→∞) |((-1)^n / x^(n+1)) / ((-1)^n / x^n)|.
Simplifying this expression, we get:
lim(n→∞) |x^n / x^(n+1)|.
Simplifying further, we get:
lim(n→∞) |x^n / (x^n * x)|.
The x^n terms cancel out:
lim(n→∞) |1 / x|.
Now, let's consider the given options and evaluate the limit for each value of x:
a) For x=2:
lim(n→∞) |1 / 2| = 1/2 < 1.
b) For x=-1/2:
lim(n→∞) |1 / (-1/2)| = 2 > 1.
c) For x=1/2:
lim(n→∞) |1 / (1/2)| = 2 > 1.
d) For x=3:
lim(n→∞) |1 / 3| = 1/3 < 1.
Based on the results, only option (a) is true, which states that for x=2, the series diverges. Therefore, the correct answer is (a).