Unit 10: Infinite sequences and series | Single Variable Calculus | Project Ulim

Math Single Variable Calculus

Unit 10: Infinite sequences and series

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About this unit

Series are sums of multiple terms. Infinite series are sums of an infinite number of terms. Don't all infinite series grow to infinity? It turns out the answer is no. Some infinite series converge to a finite value. Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in Taylor and Maclaurin series.

Defining convergent and divergent infinite series

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Convergent and divergent sequences

Worked example: sequence convergence/divergence

Partial sums intro

Partial sums: formula for nth term from partial sum

Partial sums: term value from partial sum

Working with geometric series

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Worked example: convergent geometric series

Worked example: divergent geometric series

Infinite geometric series word problem: bouncing ball

Infinite geometric series word problem: repeating decimal

Quiz 1

Answer questions from recent lessons

The nth-term test for divergence

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nth term divergence test

Practice

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Integral test for convergence

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Worked example: Integral test

Practice

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Harmonic series and p-series

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Harmonic series and 𝑝-series

Worked example: p-series

Proof of p-series convergence criteria

Practice

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Comparison tests for convergence

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Direct comparison test

Worked example: direct comparison test

Limit comparison test

Worked example: limit comparison test

Proof: harmonic series diverges

Practice

Alternating series test for convergence

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Alternating series test

Worked example: alternating series

Practice

Alternating series test

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Ratio test for convergence

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Practice

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Determining absolute or conditional convergence

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Conditional & absolute convergence

Practice

Determine absolute or conditional convergence

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Quiz 2

Answer questions from recent lessons

Alternating series error bound

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Alternating series remainder

Worked example: alternating series remainder

Practice

Alternating series remainder

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Finding Taylor polynomial approximations of functions

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Taylor & Maclaurin polynomials intro (part 1)

Taylor & Maclaurin polynomials intro (part 2)

Intuition behind Taylor approximations

Worked example: Maclaurin polynomial

Worked example: coefficient in Maclaurin polynomial

Lagrange error bound

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Taylor polynomial remainder (part 1)

Taylor polynomial remainder (part 2)

Worked example: estimating sin(0.4) using Lagrange error bound

Worked example: estimating eˣ using Lagrange error bound

Practice

Lagrange error bound

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Radius and interval of convergence of power series

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Power series intro

Worked example: interval of convergence

Practice

Interval of convergence

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Finding Taylor or Maclaurin series for a function

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Function as a geometric series

Geometric series as a function

Power series of arctan(2x)

Power series of ln(1+x³)

Maclaurin series of cos(x)

Representing functions as power series

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Integrating power series

Differentiating power series

Finding function from power series by integrating

Interval of convergence for derivative and integral

Practice

Integrate & differentiate power series

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Quiz 3

Answer questions from recent lessons

Optional videos

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Formal definition for limit of a sequence

Proving a sequence converges using the formal definition

Finite geometric series formula

Infinite geometric series formula intuition

Proof of infinite geometric series as a limit

Unit Test

Answer questions from the entire unit

Infinite series as limit of partial sums

Practice

Sequence convergence/divergence

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Partial sums intro

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Partial sums & series

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Proof of infinite geometric series formula

Practice

Infinite geometric series

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Direct comparison test

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Limit comparison test

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Worked example: coefficient in Taylor polynomial

Visualizing Taylor polynomial approximations

Practice

Taylor & Maclaurin polynomials

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Maclaurin series of sin(x)

Maclaurin series of eˣ

Worked example: power series from cos(x)

Worked example: cosine function from power series

Worked example: recognizing function from Taylor series

Visualizing Taylor series approximations

Euler's formula & Euler's identity

Geometric series interval of convergence

Practice

Function as a geometric series

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Maclaurin series of sin(x), cos(x), and eˣ

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Integrals & derivatives of functions with known power series

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Proof of p-series convergence criteria

Practice