Unit 7 of 10

Infinite Sequences and Series

  • 21 lessons
  • Parts 179 to 220

What does it mean to add infinitely many numbers, and when does the answer make sense? Sequences and series answer that question, and they are how calculators and computers evaluate functions like sine and the exponential.

You will study convergence of sequences, the main convergence tests for series, power series, and Taylor and Maclaurin series, ending with the binomial series.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Sequences

  1. Definition of a Sequence

    What a sequence is as a function on the positive integers, how explicit and recursive rules produce its terms, and how to decide whether it is bounded, positive, monotonic or alternating.

    Theory part 179Practice part 180

  2. Convergence of Sequences

    The precise epsilon N meaning of a sequence converging to a limit, the ways a sequence can fail to converge, and why every bounded monotonic sequence must converge.

    Theory part 181Practice part 182

  3. Sequence Theorems

    The rules that let you find limits of sequences without going back to epsilon each time, from limit laws and the squeeze theorem to subsequences and the Stolz-Cesàro theorem.

    Theory part 183Practice part 184

Infinite Series

  1. Series and Partial Sums

    What it means to add infinitely many numbers, how partial sums turn that question into a limit of a sequence, and how the terms, the sum and the remainder differ.

    Theory part 185Practice part 186

  2. Geometric Series

    How to add a geometric series exactly, why it converges precisely when the absolute value of the ratio is less than one, and how this turns repeating decimals into fractions.

    Theory part 187Practice part 188

  3. Telescoping and Harmonic Series

    How cancellation gives exact partial sums for telescoping series, why the harmonic series diverges even though its terms shrink to zero, and how this settles many p-series.

    Theory part 189Practice part 190

  4. Series Theorems

    The basic facts every series obeys, including the nth term test for divergence, adding convergent series, ignoring finitely many terms, and the Cauchy criterion.

    Theory part 191Practice part 192

Convergence and Convergence Tests

  1. The Integral Test

    How to decide convergence of a series of positive decreasing terms by comparing it with an improper integral, which p-series converge, and how the integral bounds the error of a partial sum.

    Theory part 193Practice part 194

  2. The Comparison Tests

    How to settle a series of nonnegative terms by comparing it with a series you already understand, either term by term or through the limit of the ratio of terms.

    Theory part 195Practice part 196

  3. Alternating Series

    Why a series whose signs alternate converges when the sizes of its terms shrink steadily to zero, and how to bound the error of stopping early by the first term left out.

    Theory part 197Practice part 198

  4. Absolute and Conditional Convergence

    The difference between series that converge because their terms are small and series that converge only through cancellation, and why only the first kind can be reordered safely.

    Theory part 199Practice part 200

  5. The Ratio and Root Tests

    How the ratio and root tests compare a series with a geometric series, when they prove absolute convergence or divergence, and why a limit of exactly one decides nothing.

    Theory part 201Practice part 202

  6. Estimates of Sums and Test Selection

    How to bound the error of a partial sum with integrals, geometric tails or the alternating estimate, how many terms that bound requires, and how to choose a convergence test quickly.

    Theory part 203Practice part 204

Power Series

  1. Definition of Power Series

    What a power series is, how its center and coefficients are read off, and how to find the radius and the exact interval where it converges, endpoints included.

    Theory part 205Practice part 206

  2. Pointwise and Uniform Convergence

    The difference between functions converging at each point and converging with one error bound everywhere, the M-test for series, and when limits pass through integrals and derivatives.

    Theory part 207Practice part 208

  3. Power Series Theorems

    How to add, multiply, divide, substitute, differentiate and integrate power series, where each result is valid, and what happens at the endpoints, including Abel's theorem.

    Theory part 209Practice part 210

Taylor and Maclaurin Series

  1. Taylor Polynomials and Taylor’s Theorem

    How to build the polynomial that matches a function and its first n derivatives at a point, and how the Lagrange and integral remainders give a guaranteed bound on its error.

    Theory part 211Practice part 212

  2. Definition of Taylor and Maclaurin Series

    What the Taylor series of a function is, what makes a function analytic, why a smooth function can fail to equal its series, and the standard exponential, trigonometric, logarithmic and geometric series.

    Theory part 213Practice part 214

  3. Taylor and Maclaurin Series Theorems

    When a Taylor series really equals its function, how Taylor's inequality proves it, why power series coefficients are unique, and which operations on series are valid where.

    Theory part 215Practice part 216

  4. Applications of Taylor and Maclaurin Series

    How Taylor series approximate values, settle indeterminate limits, compute integrals with no elementary antiderivative, estimate their own error and justify the small angle approximations used in physics.

    Theory part 217Practice part 218

  5. Binomial Series

    How the binomial theorem extends to any real exponent as an infinite series, where that series converges including the endpoints, and how the finite binomial and multinomial theorems fit in.

    Theory part 219Practice part 220