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Summary Notes for 1ZB3 Midterm 1.odt

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Chris Mc Lean

Summary Notes for 1ZB3 Midterm 1 (part one) Proof By Induction • Shows that a property holds for all elements in a list • Three steps o Base Case: Show that the property works for a number (n) o Induction: Apply a variable ( n = k ), now plug in ( k +1 ) into the ‘k’ location of the property  Rearrange everything (should probably look at lecture 4 if not sure how) o Conclusion: Works for every point of the list Infinite Sequences • Is an infinite list of elements o If a polynomial/polynomial, take the highest order of the top and bottom (the ones that are big), ignore everyone else and proceed o If top and bottom both go to infinity -> L’Hospital that shit and proceed • Take the limit o If it comes to a number, it Converges o DNE, or infinity means it Diverges Series • Are a type of sum • Take the limit (S ) approaching infinity  if it’s not this form, it means it’s m more complicated than that o Use a partial sum (plug in infinity for the ‘n’ value for the first 5 cases or so)  If it comes to a number, it Converges (or looks like it’s going to go to a number)  DNE, or infinity means it Diverges (looks like it’s doing random shit like rising or not moving, or flopping between positive and negative) Telescoping Sums • Type of sum (b nb n+1) • Compute S (mame as in a series), most terms will cancel with the telescoping property • Evaluate the limit directly Recurrence Relation Defined Sequences • When a term is defined by a function of the preceding term o an+1=1/4(a na) , a =1 • Bounded + Monotonic = Convergent o Can use Induction to show bounded & monotonic o Bounded: a
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