Given power series ∞
∑ (2x+5)^n / n4^n, determine the following.
n = 1
A. Find the center point for the series.
B. Find the radius of convergence.
C. Determine the interval of convergence
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 for the series, we can use the formula for the center of a power series:
Center = -c/b
In this case, the power series is in the form ∑ (2x+5)^n / n4^n. We can rewrite this as ∑ (2^n * (x+5)^n) / n4^n. From this form, we can see that the value of x that makes the terms of the series look like (x - a)^n, where a is the center, is x + 5.
So, the center is -5.
To find the radius of convergence, we can use the formula:
Radius = 1 / |b|
In this case, the power series is in the form ∑ (2x+5)^n / n4^n. From this form, we can see that b is 4.
So, the radius of convergence is 1/4.
To determine the interval of convergence, we need to find the values of x for which the series converges. The series will converge when the absolute value of (2x+5) is less than the radius of convergence (1/4).
So we set up the inequality:
|2x+5| < 1/4
Now we solve for x:
-1/4 < 2x+5 < 1/4
-21/8 < 2x < -19/8
-21/16 < x < -19/16
Therefore, the interval of convergence is (-21/16, -19/16).