Properties of Power Series

Term-by-term differentiation

Term-by-term integration

Algebra of series

Series solutions of differential equations

Term-by-term differentiation of a power series does what to its radius of convergence?

Differentiating the geometric series sum of x^n equals 1/(1 - x) gives which sum?

Integrating the geometric series and substituting -x gives ln(1 + x) equals which series?

The arctangent series at x = 1 gives Leibniz's formula for pi. What is the value of the series 1 - 1/3 + 1/5 - ...?

The arctangent series sum of (-1)^n x^(2n+1)/(2n+1) is obtained by integrating which function's series?

Integrating a power series term c_n x^n gives c_n/(n+1) x^(n+1). As a single number, what divides the coefficient? Answer with a number.

Differentiating c_n x^n gives n c_n x^(n-1). As a number, what is the new coefficient's factor? Answer with a word like n.

Multiplying two power series term by convolved term is called the Cauchy what? Answer with one word.

The Cauchy product of two power series is used to do what?

Airy's equation y double prime - x y = 0 is solved by power series with a coefficient recurrence. As a single letter, what is the unknown function? Answer with one lowercase letter.