1

Problem

Find the radius of convergence and the interval of convergence for the power series:

\$ ∑_(n = 0 to ∞) ((-1)^n \times x^n) / (2^(n+1))\$.

2

Step

The radius of convergence of the given power series is 2 and the interval of convergence is [-2, 2].

3

Step

The radius of convergence is half the length of the interval of convergence.

4

Formula

The formula for finding the radius of convergence is given by:

\$ R = 1 / (lim_(n \to \infty) | (a_n) / (a_(n+1)) | )\$.

5

Step

Where \$ a_n\$ is the nth term of the series.

6

Step

In this case, we have:

\$ a_n = ( (-1)^n \times x^n) / (2^(n+1)) \$

\$ | (a_n) / (a_(n+1)) | = | x/2 | \$

7

Step

Taking the limit as n approaches infinity. For the series to converge, we need:

\$ lim_(n \to \infty) | (a_n) / (a_(n+1)) | = lim_(n \to \infty) | x/2 | = | x/2 | \$

\$ | x/2 | < 1 \$

\$ -1 < x < 1 \$

8

Step

Multiplying by 2 gives us:

\$ -2 < x < 2 \$

9

Solution

So the interval of convergence is [-2, 2].

10

Sumup

Please summarize Problem, Clue, Hint, Formula, Steps and Solution

Choices

11

Choice-A

This choice is correct. The radius of convergence is found to be 2, and since the series converges at both endpoints (-2 and 2), the interval of convergence is \$−2,2\$

Correct [-2, 2]

12

Choice-B

This is an Incorrect option. This interval is wider than the one found in the solution. The radius of convergence is 2, not 3

Wrong [-3, 3]

13

Choice-C

This interval is narrower than the one found in the solution. The radius of convergence is 2, not 1.

Wrong [-1, 1]

14

Choice-D

This answer is Incorrect. The correct interval of convergence is \$−2, 2\$, which corresponds to option a

Wrong None of the above

15

Answer

Option

A

16

Sumup

Please summarize choices