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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

Find the interval of convergence for \( \sum \frac{(k + 2) x^{k+2}}{k + 3} \).

Determine the interval of convergence for the power series \( \sum \frac{(k + 2) x^{k+2}}{k + 3} \) using the Ratio Test. This solution provides a step-by-step process for applying the Ratio Test to find the radius of convergence and check endpoints.

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