Calculus 2

The binomial series extends the binomial theorem to any real power k: (1 + x)ᵏ = 1 + kx + k(k − 1)/2! x² + k(k − 1)(k − 2)/3! x³ + ⋯. When k is a whole number, the series stops. Otherwise it never ends, and it converges for |x| < 1.

You can differentiate or integrate a power series term by term, just like a polynomial. If f(x) = Σcₙxⁿ, then f′(x) = Σncₙxⁿ⁻¹ and ∫f(x) dx = C + Σcₙxⁿ⁺¹/(n + 1). The radius of convergence stays the same, but the endpoints must be checked again.

A Maclaurin series is a Taylor series centered at 0: f(x) = Σ f⁽ⁿ⁾(0)/n! xⁿ. The ones to know are the series for eˣ, sin x, cos x, 1/(1 − x), ln(1 + x) and arctan x. To get a new series, substitute into one of these, such as −x² for x in the series for eˣ.

A Taylor series is the power series Σ f⁽ⁿ⁾(a)/n! (x − a)ⁿ, built from every derivative of f at x = a. It is a Taylor polynomial that never stops. Where the series converges to f, the function equals its Taylor series; for eˣ, sin x and cos x, that is every x.

The radius of convergence R of a power series Σcₙ(x − a)ⁿ is how far from the center a the series converges. The interval of convergence runs from a − R to a + R, plus any endpoint where the series also converges. Find R with the ratio test, then test each endpoint separately.

A power series is an infinite series of the form Σcₙ(x − a)ⁿ = c₀ + c₁(x − a) + c₂(x − a)² + ⋯, like a polynomial that never ends. Many functions equal a power series; for example, 1/(1 − x) = 1 + x + x² + ⋯ when |x| < 1.

The Lagrange error bound tells you how far a Taylor polynomial can be from its function. If |f⁽ⁿ⁺¹⁾(z)| ≤ M for every z between a and x, then the error of the nth-degree Taylor polynomial is at most |Rₙ(x)| ≤ M|x − a|ⁿ⁺¹/(n + 1)!.

The alternating series error bound says that if an alternating series passes the alternating series test, stopping after n terms gives an error no bigger than the first term you left out: |S − Sₙ| ≤ bₙ₊₁. The true sum S always lies between two consecutive partial sums.

A Taylor polynomial approximates a function near a point x = a with a polynomial that matches the function’s value and its first n derivatives at a. The nth-degree Taylor polynomial is Pn(x) = f(a) + f′(a)(x − a) + f″(a)/2! (x − a)² + … + f⁽ⁿ⁾(a)/n! (x − a)ⁿ.

To choose a convergence test for a series, look at the form of its terms. Start with the divergence test. Then check for a geometric, p-series or telescoping series. Use the ratio test for factorials, the root test for nth powers, a comparison test for fractions of powers of n, and the alternating series test when signs alternate.