Find the radius of convergence and interval of convergence of the series.
There is an infinity sign on top of the summation symbol and "n=1" underneath it. To the right of the summation symbol is x^n/5n!
How would you find the radius of convergence and interval of convergence?
Explanation along the step by step process would greatly be appreciated, especially with problems dealing with factorials.
Thank you.
This looks suspiciously like 1/5 e^x which converges everywhere.
To find the radius of convergence and interval of convergence of a series, we can use the ratio test. In this case, the series is given as:
∞
Σ (x^n / (5n!))
n=1
Let's apply the ratio test to find the radius of convergence:
Step 1: Write down the ratio test.
The ratio test states that if the limit of the absolute value of the ratio of the (n+1)th term divided by the nth term approaches a limit L as n approaches infinity, then the series converges if L < 1 and diverges if L > 1.
Step 2: Calculate the ratio of consecutive terms.
Let's calculate the ratio of the (n+1)th term divided by the nth term:
((x^(n+1)) / (5(n+1)!)) / ((x^n) / (5n!))
Simplify this expression by canceling out common terms:
(x^(n+1)) / (5(n+1)!) * (5n!) / (x^n)
Simplify further:
(x^(n+1)) / (x^n) * (5n!) / (5(n+1)!)
Simplify:
x * (5n!) / (5(n+1)!), or x * 1/(n+1)
Step 3: Take the limit as n approaches infinity.
Let's calculate the limit of the ratio as n approaches infinity:
lim(n→∞) (x * 1/(n+1))
As n approaches infinity, 1/(n+1) approaches 0.
So, the limit becomes:
lim(n→∞) x * 0 = 0
Step 4: Determine the radius of convergence.
Since the limit L = 0, which is less than 1, we can conclude that the series converges for all values of x. This implies that the radius of convergence is infinity.
Step 5: Determine the interval of convergence.
Since the series converges for all values of x, the interval of convergence is (-∞, +∞), meaning the series converges for any real number x.
In summary, the radius of convergence is infinity and the interval of convergence is (-∞, +∞).