Math Problem Statement
Test the convergence of the series 1+(2÷5)x+(6÷9)x²+(14÷7)x³+...............
Solution
We are given the series:
Let's rewrite the terms to simplify them:
= 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$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Series
Convergence Tests
Ratio Test
Algebra
Formulas
Ratio Test formula: L = lim (n→∞) |(a_{n+1} * x^{n+1}) / (a_n * x^n)| = lim (n→∞) |a_{n+1} / a_n| * |x|
Theorems
Ratio Test
Suitable Grade Level
College level (Calculus 2 or Series/Sequence focused course)
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