Test the convergence of the series

Question

$\frac{3}{5.9.7}x +\frac{5}{10.12.11}x^{2}+\frac{7}{15.15.15}x^{3}+\frac{9}{20.18.19}x^{4}+........,x>0.$

08 Sep 2023

Answer :

Word Count : 321

To test the convergence of the series

$\frac{3}{5 \cdot 9 \cdot 7}x + \frac{5}{10 \cdot 12 \cdot 11}x^{2} + \frac{7}{15 \cdot 15 \cdot 15}x^{3} + \frac{9}{20 \cdot 18 \cdot 19}x^{4} + \ldots$

we can use the ratio test. The ratio test states that for a series $\sum_{n=1}^{\infty} a_n$ , the series converges absolutely if:

$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1$

Let's apply the ratio test to the given series:

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