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Mathematical Expression Editor

Exercises relating to power series and their convergence properties.

Compute the radius of convergence of the power series below.

When
computing a limit like remember the effect of orders of growth: , and so from the
perspective of the limit, the ’s in both the numerator and denominator are both
negligible.

Compute the radius of convergence of the power series below.

When computing a
limit like remember that we already know and that as well, so we expect as
well.

Compute the radius of convergence of the power series below.

Compute the interval of convergence for the power series below. The left endpoint of
the interval is ; it isis not included in the interval of convergence. The right endpoint of the interval is ; it is isis not included in the interval of convergence.

Reindex the series below:

Differentiate the series below term-by-term: (Note that the term goes away because
the derivative of a constant is zero.)

Integrate the series below term-by-term:

For each step below, apply a term-by-term operation, a multiplication by a
monomial, or a substitution to derive a new series formula from a known
formula.

Use the formula to derive a summation formula for , i.e.,

Use the formula you derived above to develop a power series expansion for
arctangent:

The radius of convergence of this last series is .

For each step below, apply a term-by-term operation, a multiplication by a
monomial, or a substitution to derive a new series formula from a known
formula.

Use the formula to determine the sum of the series below for : (Don’t
forget absolute values if you need them.)

Use the formula you just derived to determine the sum of the series

Use the formula you just derived to determine the sum of the series

Sample Quiz Questions

Find the full interval of convergence for the power series (Hints will not be revealed
until after you choose a response.)

First observe that because by virtue of l’Hospital’s rule applied to both limits on
the right-hand side.

This means that the radius equals . At the endpoint , the series
equals which diverges by the -th term divergence test because . At the endpoint , the
series equals which diverges for the same reason as the other endpoint, i.e., the terms
do not go to zero.

Find the full interval of convergence for the power series (Hints will not be revealed
until after you choose a response.)

First observe that because by virtue of l’Hospital’s rule applied to both limits on
the right-hand side.

This means that the radius equals . At the endpoint , the series
equals which diverges by direct comparison to the harmonic series, i.e., the -series
with . At the endpoint , the series equals which converges by the alternating
series test because the sign of the terms alternates and decreases to zero as
.

Find the full interval of convergence for the power series (Hints will not be revealed
until after you choose a response.)

First observe that because by virtue of l’Hospital’s rule applied to both limits on
the right-hand side.

This means that the radius equals . At the endpoint , the series
equals which diverges by direct comparison to the harmonic series, i.e., the -series
with . At the endpoint , the series equals which converges by the alternating
series test because the sign of the terms alternates and decreases to zero as
.

Find the full interval of convergence for the power series

First observe that because in the denominator tends to and This means that the
radius is infinite and the interval of convergence is .

Sample Exam Questions

For which values of does the series converge?

Find the interval of convergence of the power series below.

Find the interval of convergence of the power series .