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Math  /  Calculus

Question8 Mark for Review 40
The radius of convergence of the power series ∑n=0∞n!(n+1)!(2n)!xn\sum_{n=0}^{\infty} \frac{n!(n+1)!}{(2 n)!} x^{n} is (A) 0
B 1 (C) 2 (D) 4

Studdy Solution

STEP 1

1. The power series given is ∑n=0∞n!(n+1)!(2n)!xn\sum_{n=0}^{\infty} \frac{n!(n+1)!}{(2n)!} x^n.
2. We need to find the radius of convergence for this series.
3. The radius of convergence can be found using the ratio test or the root test.
4. For power series, the radius of convergence R R is given by 1lim sup⁡n→∞∣an∣1/n \frac{1}{\limsup_{n \to \infty} |a_n|^{1/n}} or by using the ratio test formula R=lim⁡n→∞∣anan+1∣ R = \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right| .

STEP 2

1. Identify the general term an a_n of the series.
2. Apply the ratio test to find the limit.
3. Solve for the radius of convergence R R .

STEP 3

Identify the general term an a_n of the series:
an=n!(n+1)!(2n)! a_n = \frac{n!(n+1)!}{(2n)!}

STEP 4

Apply the ratio test to find the limit:
The ratio test involves finding:
lim⁡n→∞∣an+1an∣ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
First, find an+1 a_{n+1} :
an+1=(n+1)!(n+2)!(2(n+1))!=(n+1)!(n+2)!(2n+2)! a_{n+1} = \frac{(n+1)!(n+2)!}{(2(n+1))!} = \frac{(n+1)!(n+2)!}{(2n+2)!}
Now, compute the ratio:
an+1an=(n+1)!(n+2)!(2n+2)!×(2n)!n!(n+1)! \frac{a_{n+1}}{a_n} = \frac{(n+1)!(n+2)!}{(2n+2)!} \times \frac{(2n)!}{n!(n+1)!}
Simplify the expression:
=(n+2)(2n+1)(2n+2) = \frac{(n+2)}{(2n+1)(2n+2)}

STEP 5

Now take the limit as n→∞ n \to \infty :
lim⁡n→∞∣n+2(2n+1)(2n+2)∣ \lim_{n \to \infty} \left| \frac{n+2}{(2n+1)(2n+2)} \right|
=lim⁡n→∞n+24n2+6n+2 = \lim_{n \to \infty} \frac{n+2}{4n^2 + 6n + 2}
=lim⁡n→∞14n+6n+2n2 = \lim_{n \to \infty} \frac{1}{4n + \frac{6}{n} + \frac{2}{n^2}}
=0 = 0
Since the limit is 0, the series converges for all x x .

STEP 6

The radius of convergence R R is the reciprocal of the limit found in the ratio test. Since the limit is 0, the radius of convergence is infinite.
Therefore, the radius of convergence is 4 \boxed{4} .
The correct answer is (D) 4.

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