Find-the-range-of-values-of-x-for-which-the-series-x-27-x-2-125-x-n-2n-1-3-is-absolutely-convergent-Help-

Question Number 185038 by Mastermind last updated on 16/Jan/23

Find the range of values of x for which

the series \frac{x}{27} + \frac{x^{2}}{125} + \dots + \frac{x^{n}}{{(2 n + 1)}^{3}} + \dots

is absolutely convergent .

Help!

Answered by FelipeLz last updated on 16/Jan/23

S = \underset{n = 1}{\sum^{\infty}} \frac{x^{n}}{{(2 n + 1)}^{3}}

\lim_{n \to \infty} ∣ \frac{\frac{x^{n + 1}}{{[2 (n + 1) + 1]}^{3}}}{\frac{x^{n}}{{(2 n + 1)}^{3}}} ∣ < 1

\lim_{n \to \infty} ∣ x \times {(\frac{2 n + 1}{2 n + 3})}^{3} ∣ < 1

\lim_{n \to \infty} ∣ x ∣ \times \lim_{n \to \infty} ∣ {(\frac{2 n + 1}{2 n + 3} \times \frac{n^{- 1}}{n^{- 1}})}^{3} ∣ < 1

∣ x ∣ \times \lim_{n \to \infty} ∣ {(\frac{2 + n^{- 1}}{2 + 3 n^{- 1}})}^{3} ∣ < 1

∣ x ∣ < 1

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