How do you find the radius of convergence and interval of convergence for the following?
Sum(n=1 to infinity) ((2x+5)^n) / (n(4^n))
Use the ratio test to determine whether a limit L exists
The ratio of the (n+1)st term : nth term is
|2x+5|/4 * n/(n+1)
as n->∞ that is just |2x+5|/4
the series converges if L < 1, so we need
|2x+5|/4 < 1
|2x+5| < 4
|x + 5/2| < 2
So, the radius of convergence is 2. Now just check the endpoints of the interval
-9/2 <= x <= -1/2
to see whether things converge there.
This article (as well as your text, presumably) should help:
http://tutorial.math.lamar.edu/Classes/CalcII/PowerSeries.aspx
To find the radius of convergence and interval of convergence for the given series, we will use the ratio test. The ratio test states that if the absolute value of the ratio of consecutive terms in a series approaches a limit as n approaches infinity, then the series converges absolutely if the limit is less than 1 and diverges if the limit is greater than 1.
Let's apply the ratio test to the given series:
First, calculate the absolute value of the ratio of consecutive terms:
|((2x+5)^(n+1))/(n+1)(4^(n+1))| / |((2x+5)^n)/(n(4^n))|
Simplifying, we get:
|((2x+5)^(n+1))/(n+1)(4^(n+1))| * |n(4^n)/((2x+5)^n)|
Now, simplify further by canceling out common factors:
|2x+5| / (n+1)(4/2x+5)
We need to find the limit of this expression as n approaches infinity. Let's separate the numerator and denominator terms:
(|2x+5| / 4) * (1 / (n+1)(2x+5))
As n approaches infinity, the term 1 / (n+1) approaches 0, so the expression becomes:
(|2x+5| / 4) * 0
Thus, the limit is 0.
Since the limit is less than 1, according to the ratio test, the series converges absolutely for all values of x. The radius of convergence is ∞, indicating that the series converges for all real numbers x.
As for the interval of convergence, since the series converges for all x, the interval of convergence is (-∞, +∞).