Revision

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Lecture 12

• Revision

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Lecture 12 Objectives

• Know the style of the midterm exam
• Review the course objectives through solved
  examples

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About the Midterm

• Covers Lectures 1-9 (Sheets 1-5)
• No Sample Midterm
• But a list of practice problems
• (Available on the inter/tranet)

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Lecture 1 Objectives

• Linearization (Tangent Line)
• Quadratic Approximation
• Taylor polynomials
• Sigma Notation

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Notes n 1 gives the linearization, andn 2
gives the quadratic approximation.
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Lecture 2 Objectives

• Sequences
• Explicit and Recursive Definitions
• Convergence of Sequences (Finding limn an)
  using
• Algebraic Manipulation (Division Up and Down, and
  Taking Logarithms)
• L'Hopital's Rule
• The Sandwich Theorem

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Famous Limits
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I.e If a sequence bn is sandwiched between two
sequences an and cn having a common limit L, then
bn has that same limit.
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Lecture 3 Objectives

• Geometric and Telescoping Series
• Partial Sums, Sums, Convergence
• The nth-Term (Divergence) Test.

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Geometric Series
The nth partial sum is sn a ar ar2
arn?1 a(1 ? rn)/(1 ? r) (if r ? 1)
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Telescoping Series An Example

• Find the nth partial sum.
• Is this series convergent?
• If yes, find its sum.

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The nth Term (Divergence) Test
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The nth-Term Test for Divergence

• If limn an ? 0 (or does not exist), then ?n an
  diverges.
• Caution If limn an 0, then ?n
  an may or may not converge.

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Lecture 4 Objectives

• Tests for positive series
• The Integral Test
• The Comparison Test
• The Limit Comparison Test
• The Ratio Test
• The Root Test

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The Integral Test

• Conditions on f
• Positive
• Continuous
• Decreasing

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The (Limit) Comparison Test
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The Ratio and the Root Test
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Lecture 5 Objectives

• Alternating Series Test
• Absolute Convergence and
• Conditional Convergence

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• I.e. The series has alternating signs with
  magnitudes decreasing to 0.

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Absolute and Conditional Convergence
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Summary of Convergence Tests
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Lecture 6 Objectives

• For a given power series, find
• The Radius of Convergence
• The Interval of Convergence
• The values of x where the power series converges
  absolutely/ conditionally

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To summarize
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Lecture 7 Objectives

• For a given function, use Taylor's formula, as
  well as power series manipulation, to find
• The Maclaurin Series
• The Taylor Series
• Identify Maclaurin series and find their sums

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Lecture 8 Objectives

• Error Estimation using
• The Alternating Series Estimation Theorem
• The Taylor Remainder Estimation Theorem
• Approximate Integrals using Taylor Series.

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If the series is alternating, use
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If not, use
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Lecture 9 Objectives

• Fourier Series Expansion

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Finding the Fourier Coefficients

• For the Fourier series

the coefficients can be calculated from
Note The limits 0 to 2? can be replaced by ?? to
?.
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• Thank you for listening.
• Wafik