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Summations

Learn common summation techniques: Arithmetic, Geometric, Harmonic, Telescoping. Discover ways to solve as telescoping sums and apply differentiation for |x|<1 with algebra and calculus. Explore combining known summations with differentiation and integration for effective problem-solving. Utilize bounding for easier proofs and approximations using integrals.

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Summations

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  1. Summations Arithmetic if x  1 Geometric if |x| < 1 Should know some common ones:

  2. Summations Harmonic Telescoping How to solve following as telescoping sum? ? Should know some common ones:

  3. Summations Have: if |x| < 1 Differentiate each side: if |x| < 1 if |x| < 1 Use algebra: Remember calculus? Similarly, can combine with integration. Combine known summations with differentiation, e.g.,

  4. Summations With arithmetic sum, trivial: To prove: Bounding is often easier & sufficient, e.g.,

  5. Summations 1 To prove: y=1/x y=1/(x+1) … 0 0 1 2 3 4 Can bound/approximate by integrals, e.g.,

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