Find the radius of convergence and the interval of convergence for the power series: \$ ∑_(n=0)^\infty (((-1)^n * x^n) / (2^(n+1)))\$.

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

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

The formula for finding the radius of convergence is given by

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

where \$ a_n\$ is the nth term of the series

In this case, we have:

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

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

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/2 < 1 \$

Multiplying by 2 gives us:

\$ - 2 < x < 2 \$

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

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\$

[-2, 2]

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

[-3, 3]

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

[-1, 1]

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

None of the above

Option

A

Can you summarize what you’ve understood in the above steps?