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# Determine the radius of convergence of ∑ (sin (an))zn

Determine the radius of convergence of the power series: Test for convergence at the boundary points if is finite.

If then unless . Thus, the series does not converge for if .

If then we apply Dirichlet’s test where is bounded for any (i.e., the partial sums are bounded) and the terms are monotonically decreasing with . Hence, the series converges for , which implies if .

If then for all so the series converges for all which implies .

# Determine the radius of convergence of ∑ (1 + 1/n)n2 zn

Determine the radius of convergence of the power series: Test for convergence at the boundary points if is finite.

We have Therefore, Hence, and the series converges for .

# Determine the radius of convergence of ∑ ((1*3*5*…*(2n-1)) / (2*4*6*…*(2n)))3 z

Determine the radius of convergence of the power series: Test for convergence at the boundary points if is finite.

We have Then applying the ratio test we have Hence, the radius of convergence is 1, and the series converges for .

# Determine the radius of convergence of ∑ (3n1/2 zn / n)

Determine the radius of convergence of the power series: Test for convergence at the boundary points if is finite.

Let . Then, Thus, the radius of convergence is , and the series converges for such that .

# Determine the radius of convergence of ∑ ((n!)2 / (2n!)) zn

Determine the radius of convergence of the power series: Test for convergence at the boundary points if is finite.

Let . Then we have Thus, the radius of convergence is . This diverges on the boundary points since if then the terms do not go to 0.

# Determine the radius of convergence of ∑ an2 zn

Determine the radius of convergence of the power series: Test for convergence at the boundary points if is finite.

Let . Then, for all for any . Therefore, the radius of convergence is .

# Determine the radius of convergence of ∑ (-1)n(z+1)n / (n2 + 1)

Determine the radius of convergence of the power series: Test for convergence at the boundary points if is finite.

Let . Then we have Thus, the series converges for such that ; hence, . Furthermore, the series converges on the boundary by comparison with the series .

# Determine the radius of convergence of ∑ (n! zn) / nn

Determine the radius of convergence of the power series: Test for convergence at the boundary points if is finite.

Let . Then, Thus, the series converges if which implies or the radius of convergence is . It diverges on the boundary points since the terms do not go to 0.

# Determine the radius of convergence of (1 – (-2)n)zn

Determine the radius of convergence of the power series: Test for convergence at the boundary points if is finite.

We have This is a geometric series which converges if and only if which implies . Hence, , and this converges at none of the boundary points.

# Determine the radius of convergence of ∑ (-1)n22nz2n / 2n

Determine the radius of convergence of the power series: Test for convergence at the boundary points if is finite.

This is an alternating series, and we have If then the terms are not going to 0 so the series is divergent. If then it is monotonically decreasing, so we know (by the Leibniz rule) that the series converges if and only if Therefore, the radius of convergence is , and the series converges at every boundary point .