 Test the convergence of the series 1+(2÷5)x+(6÷9)x²+(14÷7)x³+...............

We are given the series:

\[
1 + \left(\frac{2}{5}\right)x + \left(\frac{6}{9}\right)x^2 + \left(\frac{14}{7}\right)x^3 + \cdots
\]

Let's rewrite the terms to simplify them:

\[
1 + \left(\frac{2}{5}\right)x + \left(\frac{6}{9}\right)x^2 + \left(\frac{14}{7}\right)x^3 + \cdots
= 1 + \left(\frac{2}{5}\right)x + \left(\frac{2}{3}\right)x^2 + 2x^3 + \cdots
\]

We need to determine the general form of this series and test its convergence. Observe the pattern of the coefficients:
- First term: \( 1 \)
- Second term: \( \frac{2}{5} \)
- Third term: \( \frac{2}{3} \)
- Fourth term: \( 2 \)

The next step is to find the general term for the \( n \)-th term in the series. Based on observation, the general term appears to have a complex structure.

### General form

The general term for the series seems to follow the form:

\[
a_n x^n
\]

where \( a_n \) is some function of \( n \).

Once we have the general form, we can apply the **Ratio Test** to test for convergence.

### Ratio Test for Convergence
The Ratio Test is given by:

\[
L = \lim_{n \to \infty} \left| \frac{a_{n+1} x^{n+1}}{a_n x^n} \right| = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \cdot |x|
\]

If \( L < 1 \), the series converges; if \( L > 1 \), the series diverges; and if \( L = 1 \), the test is inconclusive.

To proceed further, we'd need a clear formula for the general term \( a_n \). However, based on the given pattern of coefficients, the series does not follow a simple geometric or power series, which suggests it might require deeper analysis for a specific form.

Would you like more details on determining the general term, or should I assist you further with this problem?

### Follow-up questions:
1. Would you like to explore the general term more rigorously?
2. Should I provide a refresher on the Ratio Test?
3. Do you want to explore other convergence tests like the Root Test?
4. Would you like to analyze specific values of \( x \) for convergence?
5. Are you interested in real-world applications of series convergence?

**Tip:** Convergence tests like the Ratio Test are particularly useful when dealing with series involving powers of \( x \).

Test the convergence of the series 1 + (2÷5)x + (6÷9)x² + (14÷7)x³ + ...............

Analyze the convergence of a complex series 1 + (2/5)x + (6/9)x² + (14/7)x³ + ... using the Ratio Test. Understand how to apply the Ratio Test to determine series convergence, with step-by-step explanations. Suitable for college-level students studying series and sequence convergence.

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