Unit 18: Power Series
A polynomial with infinitely many terms, well-behaved near its center and dead on arrival past a fixed distance.
5 topics · 22 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Defining a Power Series
- How it worksHow does plugging in a specific value of x turn a power series into an ordinary numerical series you can test?
- Worked exampleFind whether the power series sum of xn/n! converges at x = 2 by checking with the ratio test.
- Worked exampleFind the values of x for which the power series sum of n! xn converges.
- Common mistakeWhy doesn't a power series need to converge for every real number x just because it's an infinite sum?
- Going furtherWhat happens to a power series' convergence right at its center, no matter what its coefficients are?
Your turn1 of 2
For which values of x does the power series sum of xn / n factorial converge?Radius of Convergence
- How it worksHow does applying the ratio test directly to a power series turn convergence into an inequality x must satisfy?
- Worked exampleFind the radius of convergence for the power series sum of (x - 2)n divided by n times 3n.
- Worked exampleFind the radius of convergence for the power series sum of xn divided by n factorial.
- Common mistakeWhy doesn't a power series converge everywhere just because the function it represents is defined everywhere?
Your turn1 of 2
Find the radius of convergence for the power series sum of (x - 4)n divided by n times 5n.Interval of Convergence and the Endpoints
- How it worksHow does the ratio test go silent exactly at the two endpoints, forcing a different test to settle them?
- Worked exampleFind the full interval of convergence for the power series sum of (x + 1)n divided by √n.
- Worked exampleFind the interval of convergence for the power series sum of xn divided by n2.
- Common mistakeWhy isn't the interval of convergence always closed at both ends or open at both ends?
Your turn1 of 2
What is the full interval of convergence for the power series sum of (x + 1)n divided by √n?Representing Functions as a Power Series
- How it worksHow does substituting a new expression in for x reshape the geometric series into a power series for a new function?
- Worked exampleFind a power series for 1/(1 + x2) and state its interval of convergence.
- Worked exampleFind a power series for x/(1 - 2x) starting from the geometric series.
- Common mistakeWhy doesn't the geometric series representation of a function still hold true outside its radius of convergence?
- Going furtherWhat happens to the power series for 1/(1 - x) when x is substituted with -x3 instead?
Your turn1 of 2
Find the coefficient of x4 in the power series representation of 1/(1 + 4x2).Differentiating and Integrating a Power Series Term by Term
- How it worksHow does integrating the geometric series term by term produce a brand-new power series for ln(1 + x)?
- Worked exampleFind a power series for arctan(x) by integrating the power series for 1/(1 + x2) term by term.
- Going furtherWhat happens to a power series' radius of convergence when it's differentiated term by term?
- Common mistakeWhy isn't the interval of convergence guaranteed to stay exactly the same after integrating a power series?
Your turn1 of 2
Find the coefficient of x5 in the power series for arctan(2x), obtained by integrating the power series for 1/(1 + (2x)2) term by term.