To determine the interval of convergence for the series

$\sum \frac{(k + 2) x^{k+2}}{k + 3},$

we will use the Ratio Test, which is a standard method to find the radius of convergence for power series. Here are the steps:

Step 1: Set up the Ratio Test

The Ratio Test states that for a series $\sum a_k$, if

$\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = L,$

then:

• If $L < 1$, the series converges absolutely.
• If $L > 1$, the series diverges.
• If $L = 1$, the test is inconclusive.

In our case, $a_k = \frac{(k+2)x^{k+2}}{k+3}$.

Step 2: Compute $a_{k+1}$

To apply the Ratio Test, we need to find $a_{k+1}$: $a_{k+1} = \frac{(k+3)x^{k+3}}{k+4}.$

Step 3: Set up the ratio $\frac{a_{k+1}}{a_k}$

Now we compute $\frac{a_{k+1}}{a_k}$:

$\frac{a_{k+1}}{a_k} = \frac{\frac{(k+3)x^{k+3}}{k+4}}{\frac{(k+2)x^{k+2}}{k+3}}.$

Simplifying this fraction gives:

$\frac{a_{k+1}}{a_k} = \frac{(k+3) x^{k+3}}{(k+4) (k+2) x^{k+2}} = \frac{k+3}{k+4} \cdot x.$

Further simplification results in:

$\frac{a_{k+1}}{a_k} = \left(\frac{k+3}{k+4}\right) |x|.$

Step 4: Take the Limit as $k \to \infty$

To apply the Ratio Test, we now take the limit as (k \to \in