Taylor and Maclaurin Series

The coefficient formula

The remainder theorem

Binomial series and applications

Calculating with series

The coefficient c_n of the Taylor series of f at a equals which expression?

A Taylor series centered at a = 0 is called what?

The Taylor series of f equals f(x) exactly when which condition holds?

The binomial series for (1 + x)^k converges for |x| less than 1 for non-integer k, and at x = 1 converges when which condition holds?

The Maclaurin series of cos x is sum of (-1)^n x^(2n)/(2n)!. Which feature of the cosine's derivatives produces the even powers only?

Every derivative of e^x at 0 equals 1, so the Maclaurin coefficient c_n equals 1 over what function of n? Answer like n factorial.

The Maclaurin series of sin x begins with the term x. As a power, what is its exponent? Answer with a number.

The Lagrange remainder divides by (N + 1) factorial. As a number, how many more derivative orders than N does it involve? Answer with a number.

For the exponential's Maclaurin series, the remainder is at most e^|x| times |x|^(N+1) over (N + 1) factorial. Why does this tend to zero?

The generalized binomial coefficient binom(k, n) satisfies binom(k, n+1) equals binom(k, n) times (k - n) over (n + 1). As a number, what is the denominator? Answer with a number.