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.)

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

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

Find the full interval of convergence for the power series

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 .