Power Series

Solved Problems

Solution.

Here $$ and $$ Calculate the radius of convergence:

When $$ we have the convergent series $$ When $$ we obtain the divergent harmonic series $$ Thus, the initial series converges in the half-open interval $$

Solution.

Determine the radius and interval of convergence of the series.

If $$ we get the series

that converges by the alternating series test.

If $$ we get the divergent series:

Thus, the interval of convergence of the given series is $$

Solution.

We make the substitution: $$ The series then becomes $$ Calculate the radius of convergence:

Investigate convergence at the endpoints of the interval.

If $$ then the series

converges as a $$-series with $$

If $$ we get the alternating series

that converges by Leibniz's theorem.

Thus, the interval of convergence of the series $$ is $$ Since $$ the initial series converges for

Answer: the given series converges in the interval $$

Solution.

The $$th term of the series (starting from $$) is

Here

Determine the radius of convergence:

Now we investigate convergence of the power series at the endpoints.

If $$ we get

This series converges by the alternating series test (or Leibniz's theorem).

If $$ we have

Apply the integral test:

We see that the series $$ diverges. Therefore, the interval of convergence of the initial series is $$