First-year courses · Mathematics

Calculus II: sequences, series and polar curves

Convergence tests, geometric and p-series, power and Taylor series, radius of convergence, parametric and polar curves: Calculus II flashcards with examples.

43 cards6 topicsAbout 15 minutes a roundTypical first-year Calculus II, sequences, infinite series, power series and parametric or polar curves

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About this deck

Infinite series are the part of Calculus II that most students find hardest, mainly because there are many tests and choosing one takes practice. This deck covers sequences and their limits, partial sums, geometric, telescoping and p-series, the divergence, integral, comparison, limit comparison, ratio, root and alternating series tests, absolute and conditional convergence, power series with radius and interval of convergence, Taylor and Maclaurin series, the remainder bound, and the basics of parametric and polar curves.

Before starting, make sure you can compute limits at infinity, use L'Hospital's rule, evaluate improper integrals and differentiate quickly; the integral test and Taylor series lean directly on those skills.

For each test, learn three things: the conditions it needs, what conclusion it gives, and when it is inconclusive. When a card names a test, say those three things out loud before flipping. Later, shuffle the Convergence tests tag and, for each example series, decide which test you would try first and why. Write out the first few terms of every Taylor series card from memory; that is faster to recall in an exam than the general formula.

Watch out for these errors: thinking that terms going to 0 proves convergence (the harmonic series is the counterexample); forgetting to check the endpoints separately when finding an interval of convergence; and mixing up a sequence (a list of terms) with a series (the sum of those terms). Also, a ratio or root test result of exactly 1 tells you nothing.

Before you start: Calculus I and the integration half of Calculus II, especially limits at infinity, L'Hospital's rule and improper integrals.

All 43 cards

Cover the definitions with a sheet of paper and test yourself, or use the study modes above.

Sequences

Sequence: An ordered, infinite list of numbers a1, a2, a3, ..., usually given by a formula for the nth term or by a recurrence.Example: a_n = 1/n gives 1, 1/2, 1/3, 1/4, ...
Convergent sequence: A list of terms that approaches a single finite number L as n goes to infinity; otherwise it diverges.Example: (2n + 1)/(n + 3) approaches 2.
Monotone convergence theorem: Every sequence that is both monotonic (always increasing or always decreasing) and bounded converges.Example: a1 = sqrt(2), a(n+1) = sqrt(2 + a_n) increases and stays below 2, so it converges (to 2).
Standard sequence limits: r^n approaches 0 when |r| < 1; n^(1/n) approaches 1; (1 + 1/n)^n approaches e; n!/n^n approaches 0.Example: (0.9)^n approaches 0, while (1.1)^n grows without bound.
Squeeze theorem for sequences: If b_n <= a_n <= c_n for large n and b_n, c_n share the limit L, then a_n also approaches L.Example: -1/n <= cos(n)/n <= 1/n, so cos(n)/n approaches 0.

Series basics

Infinite series: The sum of all terms of a sequence, defined as the limit of its partial sums; it converges if that limit is a finite number.Example: 1/2 + 1/4 + 1/8 + ... converges to 1.
Partial sum: S_N, the total of the first N terms of a series. The series value is the limit of S_N as N goes to infinity.Example: For 1/2 + 1/4 + 1/8 + ..., S_3 = 7/8.
Geometric series: Sum of a r^n for n = 0 to infinity; it equals a/(1 - r) when |r| < 1 and diverges when |r| >= 1.Example: Sum of (1/3)^n from n = 0 is 1/(1 - 1/3) = 3/2.
Telescoping series: A series whose partial sums collapse because consecutive terms cancel, leaving only a few terms.Example: 1/(n(n + 1)) = 1/n - 1/(n + 1); partial sums are 1 - 1/(N + 1), so the sum is 1.
Harmonic series: The sum 1 + 1/2 + 1/3 + 1/4 + ..., which diverges even though its terms approach 0; partial sums grow roughly like ln N.Example: The first 10 terms add to about 2.93; the first million to only about 14.4.
p-series: Sum of 1/n^p from n = 1 to infinity; it converges if p > 1 and diverges if p <= 1.Example: Sum of 1/n^2 converges (to pi^2/6); sum of 1/sqrt(n) diverges.

Convergence tests

Divergence test (nth-term test): If the terms a_n do not approach 0, the series diverges. If they do approach 0, this test gives no conclusion.Example: Sum of n/(n + 1) diverges because the terms approach 1.
Integral test: If f is positive, continuous and decreasing with f(n) = a_n, then the series and the integral of f from 1 to infinity both converge or both diverge.Example: Sum of 1/(n ln n) from n = 2 diverges, since the integral gives ln(ln x), which grows without bound.
Direct comparison test: For positive terms with a_n <= b_n: if the sum of b_n converges, so does the sum of a_n; if the sum of a_n diverges, so does the sum of b_n.Example: 1/(n^2 + 1) <= 1/n^2, so sum of 1/(n^2 + 1) converges.
Limit comparison test: For positive terms, if a_n/b_n approaches a finite c > 0, then both series converge or both diverge.Example: (n + 1)/(n^3 + 2) compared with 1/n^2: the ratio approaches 1, so it converges.
Ratio test: Let L be the limit of |a(n+1)/a_n|. L < 1: absolute convergence; L > 1 (or infinite): divergence; L = 1: no conclusion. Good with factorials and powers.Example: Sum of n/2^n: ratio (n + 1)/(2n) approaches 1/2, so it converges.
Root test: Let L be the limit of |a_n|^(1/n). L < 1: absolute convergence; L > 1: divergence; L = 1: no conclusion. Good when a_n is something to the nth power.Example: Sum of (n/(2n + 1))^n: the nth root approaches 1/2, so it converges.
Alternating series test: A series with alternating signs, sum of (-1)^n b_n with b_n > 0, converges if b_n decreases and b_n approaches 0.Example: 1 - 1/2 + 1/3 - 1/4 + ... converges (its sum is ln 2).
Alternating series error bound: For a series that passes the alternating series test, the error in stopping after N terms is at most the size of the first omitted term, b(N+1).Example: Using 4 terms of 1 - 1/2 + 1/3 - ..., the error is at most 1/5.
Absolute convergence: A series converges absolutely when the sum of the absolute values |a_n| converges; this always implies the original series converges.Example: Sum of (-1)^n/n^2 converges absolutely because the sum of 1/n^2 converges.
Conditional convergence: A series that converges, but whose series of absolute values diverges.Example: The alternating harmonic series converges, but the harmonic series diverges.
Choosing a convergence test: Check the divergence test first; use geometric or p-series forms if visible; ratio test for factorials and exponentials; comparison for rational expressions; alternating test for sign changes.Example: Sum of 3^n/n!: factorials suggest the ratio test, which gives 0, so it converges.

Power series

Power series: A series of the form sum of c_n (x - a)^n, centred at a, whose value depends on x.Example: Sum of x^n = 1 + x + x^2 + ... is centred at 0.
Radius of convergence: The number R such that a power series converges absolutely for |x - a| < R and diverges for |x - a| > R. R can be 0 or infinity.Example: Sum of x^n/n: ratio test gives |x|, so R = 1.
Interval of convergence: All x for which a power series converges: the open interval from the radius plus whichever endpoints pass a separate test.Example: Sum of x^n/n converges on [-1, 1): at x = -1 it alternates, at x = 1 it is harmonic.
Extreme radii: A power series may converge only at its centre (R = 0) or for every real x (R = infinity).Example: Sum of x^n/n! has R = infinity; sum of n! x^n has R = 0.
Term-by-term differentiation and integration: Inside its radius of convergence, a power series can be differentiated or integrated term by term, and the radius stays the same.Example: Differentiating 1/(1 - x) = sum of x^n gives 1/(1 - x)^2 = sum of n x^(n - 1) for |x| < 1.
Building series from the geometric series: Substitute into 1/(1 - u) = sum of u^n (valid for |u| < 1) to get series for related functions.Example: 1/(1 + x^2) = sum of (-1)^n x^(2n) for |x| < 1; integrating gives the arctan x series.

Taylor series

Taylor series: The power series of f centred at a: sum of f^(n)(a)/n! times (x - a)^n, built from all the derivatives of f at a.Example: For f(x) = 1/x at a = 1: 1 - (x - 1) + (x - 1)^2 - ...
Maclaurin series: A Taylor series centred at 0: sum of f^(n)(0) x^n/n!.Example: The Maclaurin series of e^x is 1 + x + x^2/2! + x^3/3! + ...
Series for e^x: e^x = sum of x^n/n! for n = 0 to infinity, valid for every real x.Example: e is about 1 + 1 + 1/2 + 1/6 + 1/24 = 2.7083 (true value 2.71828).
Series for sin x and cos x: sin x = x - x^3/3! + x^5/5! - ... (odd powers) and cos x = 1 - x^2/2! + x^4/4! - ... (even powers), valid for all x.Example: sin(0.1) is about 0.1 - 0.001/6 = 0.099833.
Series for ln(1 + x): ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + ..., valid for -1 < x <= 1.Example: Setting x = 1 gives ln 2 = 1 - 1/2 + 1/3 - ...
Binomial series: (1 + x)^k = 1 + kx + k(k - 1)x^2/2! + ... for |x| < 1, valid for any real k.Example: sqrt(1 + x) is about 1 + x/2 - x^2/8.
Taylor polynomial: The nth-degree partial sum T_n of a Taylor series; it matches f and its first n derivatives at the centre and approximates f nearby.Example: T_2 for cos x at 0 is 1 - x^2/2; cos(0.1) is about 0.995 (true 0.995004).
Taylor remainder estimate: If |f^(n+1)| <= M between a and x, then the error |f(x) - T_n(x)| is at most M |x - a|^(n+1)/(n + 1)!.Example: e^0.1 with T_2 = 1.105: M = e^0.1 < 1.2, so error < 1.2(0.1)^3/6 = 0.0002.
Limits using series: Replace functions by the first few terms of their series to evaluate tricky limits without repeated L'Hospital.Example: (sin x - x)/x^3 = (-x^3/6 + ...)/x^3, which approaches -1/6.

Parametric and polar curves

Parametric curve: A curve traced by x = f(t) and y = g(t) as the parameter t varies, which also records direction and speed of travel.Example: x = cos t, y = sin t for 0 <= t <= 2 pi traces the unit circle counterclockwise.
Slope of a parametric curve: dy/dx = (dy/dt)/(dx/dt), wherever dx/dt is not 0.Example: x = t^2, y = t^3: dy/dx = 3t^2/(2t) = 3t/2, which is 3 at t = 2.
Parametric arc length: Length = integral of sqrt((dx/dt)^2 + (dy/dt)^2) dt over the parameter interval, if the curve is traced once.Example: x = r cos t, y = r sin t on [0, 2 pi]: integral of r dt = 2 pi r.
Polar coordinates: A point given by distance r from the origin and angle theta from the positive x-axis; x = r cos(theta), y = r sin(theta), r^2 = x^2 + y^2.Example: (r, theta) = (2, pi/3) is the point (1, sqrt(3)).
Area in polar coordinates: Area swept out by r = f(theta) from theta = a to b is (1/2) times the integral of r^2 d(theta).Example: r = 2 for theta from 0 to 2 pi: (1/2)(4)(2 pi) = 4 pi.
Converting a polar equation: Multiply or rearrange so you can substitute x = r cos(theta), y = r sin(theta) and r^2 = x^2 + y^2.Example: r = 2 cos(theta): r^2 = 2r cos(theta), so x^2 + y^2 = 2x, a circle with centre (1, 0) and radius 1.