Question

If \(R\) denotes the radius of convergence of the power series

• A. \(9R=4\)
• B. \(4R=9\)
• C. \(R=1\)
• D. \(2R=1\)

Hint:

For power series, the ratio test is often the fastest method to find the radius of convergence: \[ R=\lim_{n\to\infty}\left|\frac{a_n}{a_{n+1}}\right| \] when the limit exists.

Answer 1

The Correct Option is B

Step 1: Spot the pattern in the coefficients.
The series is $\sum a_n x^n$ with $a_n=\left(\dfrac{2\cdot4\cdots 2n}{2\cdot5\cdots(3n-1)}\right)^2$. Each new term multiplies the top by $2n+2$ and the bottom by $3n+2$.

Step 2: Pick the ratio test.
The radius is $R=\lim_{n\to\infty}\left|\dfrac{a_n}{a_{n+1}}\right|$. Because of the square, this is the square of the ratio of the inner fractions.

Step 3: Write the ratio.
The extra factors give \[ \frac{a_n}{a_{n+1}}=\left(\frac{3n+2}{2n+2}\right)^2. \]

Step 4: Take the limit.
For large $n$ the leading terms decide things, so $\dfrac{3n+2}{2n+2}\to\dfrac32$. Squaring, \[ R=\left(\frac32\right)^2=\frac94. \]

Step 5: Match the option.
From $R=\dfrac94$ we get $4R=9$.
\[ \boxed{4R=9} \]