In the following series x is a real number. In each case, use the ratio test to determine the radius of convergence of the series. Analyze the behavior at the endpoints in order to determine the interval of convergence.
a. Summation n=0 to infinity (n(x^n))/(n^2 + 2)
b. Summation n=1 to infinity [(n^2)(x-2)^n]/(2^n)
c. Summation n=1 to infinity [(4^n)(x^n)]/(n^2)
To determine the radius of convergence of each series using the ratio test, we need to compare the absolute value of the ratio of consecutive terms with 1. Let's start with the given series:
a. Summation n=0 to infinity (n(x^n))/(n^2 + 2)
The ratio of consecutive terms is given by (n+1)(x^(n+1))/(n^2 + 2) * (n^2 + 2)/(n(x^n)).
Taking the absolute value of this ratio, we have |(n+1)(x^(n+1))/(n^2 + 2) * (n^2 + 2)/(n(x^n))|.
Simplifying this expression, we get |(n+1)x(n+1)/(nx^n)|.
Since we want to determine the radius of convergence, we need to find the limit as n approaches infinity of this expression:
lim n→∞ |(n+1)x(n+1)/(nx^n)|.
Applying the ratio test, we know that if this limit is less than 1, the series converges, and if it is greater than 1, the series diverges. If the limit equals 1, the test is inconclusive.
Now let's analyze the behavior at the endpoints to determine the interval of convergence:
For the series a, when x = -1, the series becomes Summation n=0 to infinity (-1)^n/(n^2 + 2).
To determine the convergence or divergence at this endpoint, we can use the alternating series test. We check if the terms alternate in sign and if the absolute values of the terms decrease as n increases.
For n > 0, the terms (-1)^n/(n^2 + 2) alternate in sign and decrease in absolute value. Moreover, the term for n = 0 is positive. Therefore, the series converges when x = -1.
Similarly, we can analyze the behavior at the other endpoint x = 1.
By applying the ratio test and analyzing the behavior at the endpoints, we can determine the radius of convergence and the interval of convergence for each series.
To determine the radius of convergence of a series and analyze its behavior at the endpoints, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms in a series exists and is less than 1, then the series converges. On the other hand, if the limit is greater than 1 or does not exist, the series diverges.
a. For the series ∑(n(x^n))/(n^2 + 2), let's apply the ratio test. Take the absolute value of the ratio between consecutive terms and find the limit as n approaches infinity:
|r_n+1/r_n| = |(n+1)(x^(n+1))/((n+1)^2 + 2) * (n^2 + 2)/(nx^n)|
Simplifying gives: |(n+1)x|/|n+1|x^n| = |x| * (n+1)/n
Taking the limit: lim as n approaches infinity |x| * (n+1)/n
This limit simplifies to |x|.
Since |x| is just a constant and not dependent on n, the limit is independent of n and hence doesn't approach 0, satisfying the condition for the ratio test. Therefore, the series converges for any x within a certain interval.
Now, to determine the interval of convergence and behavior at the endpoints, we need to consider the convergence when |x| = 1.
When x = 1, the series becomes ∑(n/n^2 + 2). This is a unique series that converges, so the series converges at x = 1.
When x = -1, the series becomes ∑((-1)^n * n/n^2 + 2), which is the alternating harmonic series. This series converges by the Alternating Series Test, so the series converges at x = -1.
Hence, the interval of convergence for part (a) is -1 ≤ x ≤ 1.
b. For the series ∑((n^2)(x-2)^n)/(2^n), let's apply the ratio test. Take the absolute value of the ratio between consecutive terms and find the limit as n approaches infinity:
|r_n+1/r_n| = |(n+1)^2(x-2)^(n+1))/(2(n^2)(x-2)^n)|
Simplifying gives: (n+1)^2/2 * |x-2|/n^2
Taking the limit: lim as n approaches infinity (n+1)^2/2 * |x-2|/n^2
This limit simplifies to |x-2|/2.
Again, this is a constant that doesn't depend on n, and its value is not 0. Therefore, the series converges for any x within a certain interval.
Now, let's consider the convergence when |x-2| = 2, which gives us x = 0 and x = 4.
When x = 0, the series becomes ∑(n^2/2^n), which converges by the Ratio Test. Hence, the series converges at x = 0.
When x = 4, the series becomes ∑(n^2/2^n), which again converges. Thus, the series converges at x = 4.
Therefore, the interval of convergence for part (b) is 0 ≤ x ≤ 4.
c. For the series ∑((4^n)(x^n))/n^2, let's apply the ratio test. Take the absolute value of the ratio between consecutive terms and find the limit as n approaches infinity:
|r_n+1/r_n| = |(4^(n+1))(x^(n+1))/((n+1)^2) * (n^2)/(4^n)(x^n)|
Simplifying gives: 4|x| * (n^2)/(n+1)^2
Taking the limit: lim as n approaches infinity 4|x| * (n^2)/(n+1)^2
This limit simplifies to 4|x|.
Again, this is a constant that doesn't depend on n and is not equal to 0. Therefore, the series converges for any x within a certain interval.
Now, let's consider the convergence when |4x| = 4, which gives us x = 1 and x = -1.
When x = 1, the series becomes ∑(4^n/n^2), which converges by the Ratio Test. Hence, the series converges at x = 1.
When x = -1, the series becomes ∑((-1)^n * 4^n/n^2), which alternates between positive and negative values and satisfies the conditions of the Alternating Series Test. Therefore, the series converges at x = -1.
Thus, the interval of convergence for part (c) is -1 ≤ x ≤ 1.
To summarize:
a. The series ∑(n(x^n)/(n^2 + 2) converges for -1 ≤ x ≤ 1.
b. The series ∑((n^2)(x-2)^n)/(2^n) converges for 0 ≤ x ≤ 4.
c. The series ∑((4^n)(x^n))/n^2 converges for -1 ≤ x ≤ 1.