Question

Power SeriesIn Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.∑ (from n = 1 to ∞) (x + 4)ⁿ/(n3ⁿ)

Accepted Answer

Identify the given power series: $$\displaystyle \sum_{n=1}^{\infty} \frac{(x+4)^n}{n 3^n}. $$Notice that this is a power series centered at $$x = -4$$ because the term is $$(x+4)^n. To $$find the radius of convergence, apply the Root Test or Ratio Test. The Ratio Test is often easier here. Consider the general term $$a_n = \frac{(x+4)^n}{n 3^n}. $$Compute the limit $$L = \lim_{n 	o \infty} \left| \frac{a_{n+1}}{a_n} ight|. $$Calculate the ratio inside the limit: $$\left| \frac{a_{n+1}}{a_n} ight| = \left| \frac{(x+4$$)^{n+1}$$}{(n+1) 3$$^{n+1}$$} \cdot \frac{n 3^n}{(x+4)^n} ight| = \left| \frac{(x+4)}{3} \cdot \frac{n}{n+1} ight|. As$$ $$n 	o \infty$$, $$\frac{n}{n+1} 	o 1$$, so the limit simplifies to $$L = \left| \frac{x+4}{3} ight|. $$The Ratio Test tells us the series converges if $$L \u003c 1$$, so the radius of convergence $$R$$ satisfies $$\left| x+4 ight| \u003c 3. $$This means the radius of convergence is $$R = 3$$, and the interval of convergence is $$(-4 - 3, -4 + 3) or$$ $$(-7, -1)$$, but we still need to check the endpoints $$x = -7$$ and $$x = -1$$ separately. To determine convergence at the endpoints, substitute $$x = -7$$ and $$x = -1$$ into the series and analyze the resulting series. For each endpoint, check if the series converges absolutely (by testing the absolute value of terms) or conditionally (converges but not absolutely). This often involves comparing to known series like the p-series or alternating series tests.

Power Series
In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.

∑ (from n = 1 to ∞) (x + 4)ⁿ/(n3ⁿ)

Risposta video verificata per un problema simile:

Concetti chiave

Radius and Interval of Convergence

Absolute Convergence

Conditional Convergence