Calculus II Midterm #3 Review

• §11.1: Sequences
  • Sequences as functions using the natural numbers as the domain
  • Recursive definition (a_n+1 in terms of a_n) vs general definition (a_n in terms of n)
  • Arithmetic sequences vs geometric sequences
  • Formal definition of limit of a sequence (both finite and infinite)
  • Monotonic sequences, bounded sequences, Monotonic Sequence Theorem


• §11.2: Series
  • Finite vs infinite, convergent vs divergent
  • Arithmetic series, geometric series, harmonic series
  • Partial sums; formulas for arithmetic and geometric series
  • Limit Test for Divergence
  • Telescoping sums


• §11.3: Integral Test
  • Rewrite sum of a sequence as integral (from 1 to ∞) of a function
  • If improper integral converges, so does series
  • If improper integral diverges, so does series
  • Useful for any sequence whose general term is easily integrable


• §11.4: Comparison Tests
  • Comparison Test:
    • If it's always less than the terms of a convergent series, then it also converges.
    • If it's always greater than the terms of a divergent series, then it also diverges.
    • BEWARE: "less than divergent" or "more than convergent" implies nothing!
  • Limit Comparison Test:
    • If the limit of the ratio of two sequences is nonzero and finite,
    • then the related series must either both converge or both diverge.
  • These tests can be useful for analyzing a series that resembles an already known series.


• §11.5: Alternating Series
  • Absolute convergence vs conditional convergence
  • Alternating Series Test
  • Estimating with partial sum; upper bound on error (remainder)


• §11.6: Ratio Test
  • If the limit L of the ratio | a_n+1 / a_n | is...
    • ...less than 1, the series converges.
    • ...greater than 1, the series diverges.
    • ...equal to 1, try another test.


• §11.7: Strategy for Series Testing
  • Ultimately this is about developing an intuition, and intuition comes with practice.
  • Try to describe the series by its form, and consider what test(s) would be most useful.
  • For more details and examples, see pp. 721-722 of the textbook.


• §11.8: Power Series
  • Interval of convergence, center of convergence, radius of convergence
  • Interval of convergence may also be a single number or all real numbers.
  • To find interval of convergence, use Ratio Test (set L<1) and solve for x.
  • Ratio Test will fail (L=1) at endpoints, so check them using another test.


• §11.9: Functions Represented by Power Series
  • Series can be integrated (and differentiated) term-by-term.
  • This makes it easy to integrate functions that can be written as power series.
  • Also: power series representations for 1/(1-x), tan^-1(x), and ln(1-x)


• §11.10: Maclaurin Series & Taylor Series
  • ANY infinitely differentiable function can be approximated as a power series!
  • Taylor Series: f(x) = f(a)/0! + (x-a)·f'(a)/1! + (x-a)²·f''(a)/2! + (x-a)³·f'''(a)/3! + ...
  • ...where a is the "center"; partial sum approximations will be most accurate near x=a.
  • Maclaurin Series: special case; same as above, but centered at a=0.
  • Maclaurin is easier, but inaccurate far from 0 and impossible if f(0) doesn't exist.
  • Know the Maclaurin series for e^x, sin(x), and cos(x), or be able to find them.


• Also: Complex Exponents!
  • If imaginary number x=iθ is used as input in e^x, the power series "unzips" into...
  • Euler's Formula: e^iθ = cos θ + i·sin θ     (and Euler's Identity: e^πi = -1)
  • With the Product Rule, this becomes e^a+bi = e^a·e^bi = e^a·(cos b + i·sin b)
  • For more details and practice problems, see pp. A63-A64 near the end of the textbook.

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