Outline: sequences and series

Sequences

• Know what it means for a sequence to:
  • converge
  • diverge
  • converge to a real number $L$
  • diverge to $\infty$
• Determine whether a sequence converges (and if so, what its limit is) using:
  • definition of "convergent" (respectively, of "limit") for sequences
  • familiarity with limits of common functions of $n$ such as powers of $n$, $e^n$, $\ln(n)$, etc.
  • L'Hospital's rule
  • rules for $\lim_{n\to \infty} (a_n + b_n)$, $\lim_{n \to \infty} a_n b_n$, $\lim_{n \to \infty} f(a_n)$ where $f$ is a continuous function, etc.
  • every bounded, monotonic sequence is convergent
  • changing finitely many terms does not affect convergence, or affect the limit if it exists

Series

• Know the definition of the $n^\text{th}$ partial sum of a series
• Know the difference between the sequence of terms of a series and the sequence of partial sums of that series
• Know what it means for a series to converge, diverge, converge to $L$, or diverge to $\infty$ in terms of the sequence of partial sums
• Determine whether a series converges (and if so, what its sum is) using:
  • definition of "convergent" (respectively, of "sum") for series
  • re-writing as a telescoping series
  • rules for $\sum_{n=1}^\infty (a_n + b_n)$, etc.
  • changing finitely many terms does not affect convergence—know how it affects the sum
• Determine whether a series converges using the following tests (know when the tests apply, and what they say when they do):
  • the "test for divergence"
  • integral test
  • comparison test
  • limit comparison test
  • alternating series test
  • ratio test
  • root test
• Be familiar with the definitions of, and behavior of, the following examples:
  • geometric series
  • $p$-series (including harmonic series)
  • alternating harmonic series
• Know the alternating series estimation theorem
• Absolute vs. conditional convergence
  • know the definition of absolute convergence and of conditional convergence
  • understand that absolute convergence implies convergence (this is not immediate from the definitions)
  • know an example of conditional convergence

Power series

• Know what a power series in $x-a$ (also called a power series centered at $a$) is
• Interval of convergence
  • know what it means for $R$ to be the radius of convergence of a power series
  • every power series has a radius of convergence, possibly zero or $\infty$ (this is not immediate from the definitions)
  • know what the interval of convergence of a power series means
  • find the interval of convergence by finding the radius of convergence and testing the endpoints if applicable
• Differentiation and integration of power series can be done term-by-term and does not change the radius of convergence
• Representing functions as (sums of) power series
  • relate to a function with a known power series, for example that of $1/(1-x)$ or $(1+x)^k$, by substitution, algebra, calculus, etc.
  • calculate the Taylor series coefficients using the derivatives of the function
• Taylor series
  • know the definition of the Taylor series of a function $f(x)$ at a point $a$
  • The Maclaurin series is the Taylor series at $a = 0$
  • know Maclaurin series for common functions $e^x$, $\sin x$, etc. or be able to derive them
• Taylor polynomials
  • know the definition of the $n^\text{th}$ degree Taylor polynomial of a function $f(x)$ at a point $a$
  • understand the motivation: this allows us to approximate the values of $f$ near $a$
  • calculate the first few Taylor polynomials of a product $f(x)g(x)$ in terms of power series for $f(x)$ and $g(x)$