Series & Convergence
A copy-able sheet — Maclaurin expansions with their intervals, every convergence test, and the closed-form sums.
Showing 30 of 30. Click any formula to copy it.
C(k, n) is the generalised binomial coefficient k(k − 1)…(k − n + 1) ÷ n!, which is why the binomial series does not terminate unless k is a non-negative integer. Every interval of convergence above is stated for real x.
A Maclaurin series rewrites a function as an infinite polynomial. The five worth memorising are eˣ, sin x, cos x, ln(1 + x), and 1 ÷ (1 − x). The first three converge for every x; the last two only on an interval — 1 ÷ (1 − x) = 1 + x + x² + … needs |x| < 1.
Why a function becomes a polynomial
A Taylor series is built from derivatives at a single point: each term matches one more derivative of the original function there, so the partial sums hug the curve ever more tightly near that point. A Maclaurin series is the same construction centred at zero, which is why it is the version tabulated — most standard functions are cleanest there.
The catch is that matching derivatives at one point does not guarantee agreement everywhere. The interval of convergence says how far the polynomial can be trusted, and it is as much a part of the formula as the terms themselves. Writing 1 ÷ (1 − x) = Σxⁿ without the condition |x| < 1 gives nonsense at x = 2, where the left side is −1 and the right side diverges.
Derive rather than memorise
Most expansions on the sheet follow from a handful by substitution. Replacing x with −x² in the geometric series gives the series for 1 ÷ (1 + x²), and integrating that term by term gives the arctangent series. Differentiating the sine series gives the cosine series. Four or five memorised expansions plus these moves cover nearly everything a course asks for.
The Maclaurin series worth knowing
Every interval below is stated for real x. Outside it the series says nothing about the function.
| Function | Series | Converges for |
|---|---|---|
| eˣ | 1 + x + x²/2! + x³/3! + … | All x |
| sin x | x − x³/3! + x⁵/5! − … | All x |
| cos x | 1 − x²/2! + x⁴/4! − … | All x |
| 1 ÷ (1 − x) | 1 + x + x² + x³ + … | |x| < 1 |
| ln(1 + x) | x − x²/2 + x³/3 − … | −1 < x ≤ 1 |
| arctan x | x − x³/3 + x⁵/5 − … | |x| ≤ 1 |
| (1 + x)ᵏ | 1 + kx + k(k − 1)x²/2! + … | |x| < 1 |
Choosing a convergence test
Work down the list — the cheap tests rule out most series before the expensive ones are needed.
| If the series has… | Use | Note |
|---|---|---|
| Terms that do not approach 0 | nth-term test | Proves divergence only, never convergence |
| A constant ratio between terms | Geometric test | Converges when |r| < 1, to a ÷ (1 − r) |
| The form 1 ÷ nᵖ | p-series test | Converges when p > 1; p = 1 is the harmonic series |
| Factorials or nth powers | Ratio test | Inconclusive when the limit equals 1 |
| n appearing as an exponent | Root test | Often easier than the ratio test here |
| A term you can integrate | Integral test | Needs positive, decreasing, continuous terms |
| Alternating signs | Alternating series test | Terms must decrease monotonically to 0 |
| A resemblance to a known series | Comparison or limit comparison | Compare against a p-series or geometric series |
The traps in convergence questions
The nth-term test is the one most often misused. If the terms approach zero, the test says nothing at all — the harmonic series Σ1 ÷ n has terms going to zero and still diverges. The test can only ever prove divergence.
Two more are worth naming. Absolute and conditional convergence are different things: Σ(−1)ⁿ ÷ n converges, but Σ1 ÷ n does not, so that series is conditionally convergent and its terms can be rearranged to sum to any value you like. And the endpoints of an interval of convergence have to be tested separately — the ratio test gives the radius but is always inconclusive exactly at the ends, which is why ln(1 + x) converges at x = 1 but not at x = −1.