This might be easier to understand:
Given power series:
∞
∑ (2x+5)^n / n4^n
n = 1
- determine the following.
A. Find the center point for the series.
B. Find the radius of convergence.
C. Determine the interval of convergence
To find the center point of the series, we look at the term inside the parentheses: 2x + 5.
A. Find the center point for the series:
The center point is the value of x that makes 2x + 5 equal to zero. So, we solve the equation 2x + 5 = 0:
2x + 5 = 0
2x = -5
x = -5/2
Therefore, the center point of the series is x = -5/2.
B. Find the radius of convergence:
The radius of convergence is determined by the distance from the center point to the closest point where the series converges. To find the radius, we need to determine the distance between the center point and the nearest singularity or endpoint.
In this case, as the exponent n in the series increases, (2x + 5)^n term becomes larger, so we need to make sure that this term does not grow indefinitely. To ensure convergence, we consider the ratio test:
lim(n->∞) |[(2x+5)^(n+1) / n^4^(n+1)] / [(2x+5)^n / n^4^n]|
= lim(n->∞) |(2x+5)^(n+1) / (2x+5)^n| * |n^4^n / n^4^(n+1)|
= lim(n->∞) |(2x+5) / n| * |n^4^n / (2x+5)*n^4^n|
To check for convergence, we take the absolute value of the expression above, and find the limit as n approaches infinity:
lim(n->∞) |(2x+5) / n| * |n^4^n / (2x+5)*n^4^n| = |2x+5 / (2x+5)| * |1/n|
= 1 * |1/n|
= 1/n
The limit of 1/n as n approaches infinity is 0. Therefore, the ratio test is inconclusive and doesn't provide any information regarding the convergence of the series.
Instead, we can examine the intervals of convergence by considering the values of x where the series converges.
C. Determine the interval of convergence:
To determine the interval of convergence, we consider the endpoints and check if the series converges or diverges at those points.
First, we evaluate the series at the center point, x = -5/2. Substituting this value into the series:
∑ ((2*(-5/2) + 5)^n / n^4^n)
= ∑ ((-5)^n / n^4^n)
This series is an alternating series, as the sign changes as n increases. Furthermore, the terms decrease in absolute value as n increases. Therefore, by the Alternating Series Test, this series converges when evaluated at x = -5/2.
Next, we examine the endpoints of the interval.
At the left endpoint, x = -∞, the series becomes:
∑ (2*(-∞) + 5)^n / n^4^n
= ∑ (-∞)^n / n^4^n
As (-∞)^n is not well-defined when n is even, this series does not converge at the left endpoint x = -∞.
At the right endpoint, x = +∞, the series becomes:
∑ (2*(+∞) + 5)^n / n^4^n
= ∑ (+∞)^n / n^4^n
This series diverges at the right endpoint x = +∞.
Therefore, we can conclude that the interval of convergence for the given series is (-∞, -5/2].
To find the center point, radius of convergence, and interval of convergence of the power series, we can use the ratio test.
The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, the series converges. Conversely, if the limit is greater than 1, the series diverges.
Let's apply the ratio test to the given series:
∞
∑ (2x+5)^n / n4^n
n = 1
Step 1: Calculate the limit using the ratio test:
lim(n→∞) |[(2x+5)^(n+1) / (n+1)4^(n+1)] / [(2x+5)^n / n4^n]|
Step 2: Simplify the expression by canceling out common terms:
lim(n→∞) |(2x+5)^(n+1) / (n+1)4^(n+1) * n4^n / (2x+5)^n|
Step 3: Simplify further by canceling out (2x+5)^n terms:
lim(n→∞) |(2x+5) / (n+1)4 * n4|
Step 4: Evaluate the limit:
lim(n→∞) |(2x+5) / (n+1)4 * n4|
= |(2x+5) / 4|
Step 5: Set the absolute value of the ratio less than 1 to determine the radius of convergence:
|(2x+5) / 4| < 1
Step 6: Solve for x to find the interval of convergence:
-(2x+5) / 4 < 1 and (2x+5) / 4 < 1
Solving the first inequality:
-(2x+5) / 4 < 1
2x+5 > -4
2x > -9
x > -9/2
Solving the second inequality:
(2x+5) / 4 < 1
2x+5 < 4
2x < -1
x < -1/2
Thus, the interval of convergence is -9/2 < x < -1/2.
Finally, to find the center point, we take the average of the two endpoints of the interval of convergence:
Center Point = (-9/2 + (-1/2)) / 2
= -10/2
= -5
Therefore, the center point is -5, the radius of convergence is 4, and the interval of convergence is -9/2 < x < -1/2.