Calculus 2, part 2 of 2: Sequences and series
Single variable Calculus: sequences and series of numbers and of real-valued functions of one real variable
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60.5 hours
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Apr 2025
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What you will learn
How to solve problems concerning sequences and series (illustrated with 378 solved problems) and why these methods work.
Various ways of defining number sequences: with help of a descriptive definition, an explicit (closed) formula, a recursive definition.
Limits of number sequences, both finite and infinite (repetition and continuation of the topic introduced in Calc1p1).
Arithmetic on extended reals (repetition and continuation of the topic introduced in Calc1p1).
Dealing with indeterminate forms (repetition and continuation of the topic introduced in Calc1p1).
Standard limits, comparing infinities (repetition and continuation of the topic introduced in Calc1p1).
Squeeze Theorem for sequences (repetition and continuation of the topic introduced in Calc1p1).
Weierstrass' Theorem for sequences (repetition and continuation of the topic introduced in Calc1p1).
Functions defined for all positive arguments and their role in examining number sequences (monotonicity, limits).
Solving recurrence relations: closed formula for linear recurrences of order two (with or without initial conditions).
Stolz-Cesàro Theorem for computing limits of indeterminate forms.
Ratio test for sequences for determining convergence or divergence.
Cauchy Theorem (convergence of the arithmetic means of a convergent sequence).
AM-GM inequality; convergence of the geometric means of a convergent sequence.
Examining the differences of consecutive terms of the sequence (a_n) for determining convergence of a_n/n.
Examining the quotients of consecutive terms of the sequence (a_n) for determining convergence of the n-th roots of a_n.
Some applications of sequences (discretisation, approximations); sequences in other courses.
Cauchy sequences and completeness of the set of real numbers; Bolzano-Weirestrass' Theorem.
Series as limits of partial sums of sequences: definition and examples.
Convergent and divergent series: simple and advanced examples.
Arithmetic and geometric series and their convergence.
p-series, their relation to p-integrals; discussion of their convergence.
Comparison criteria for series.
Integral test for series.
Ratio test for series.
Root test for series.
Sequences of real-valued functions (a very brief introduction).
Series of real-valued functions (a very brief introduction).
Power series and their radius of convergence.
Taylor polynomials and Taylor series, smooth functions.
Applications of Taylor polynomials for approximating values of functions.
Applications of Taylor polynomials for computing limits of indeterminate forms.
Sneak peek into the content of the next course (Real Analysis: metric spaces).
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udemy ID
9/25/2024
course created date
4/12/2025
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