Practice Power Series in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
An infinite series of the form \sum_{n=0}^{\infty} a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + \cdots where c is the center and a_n are the coefficients. A power series defines a function of x wherever it converges.
A power series is an 'infinite polynomial' centered at c. For each value of x, you get a number series that may or may not converge. The set of x-values where it converges forms an interval centered at c, and within that interval, the power series behaves like a well-defined function.
Example 1
mediumFind the radius of convergence of \displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n+1}.
Example 2
hardFind the interval of convergence of \displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n}.
Example 3
easyFind the radius of convergence of \displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n!}.
Example 4
mediumDifferentiate \sum_{n=0}^{\infty} x^n = \frac{1}{1-x} term by term to find a new series identity.