Calculus: L'Hôpital's Rule, Sequences, and Series
Explore the concepts of L'Hôpital's Rule, convergence of sequences, and geometric series in Calculus. Understand how to compute limits, find the convergence of sequences, and work with series in mathematics.
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Lecture 5: LHpitals Rule, Sequences, and Series
Indeterminate forms ?(?) ?(?)when ? ? = How do we compute lim ? ? ? ? = 0? ?(?) ?(?)when Similarly, how do we compute lim ? lim ? ? ? = and lim ? ? ? = ?
Computing Indeterminate forms If ? ? = ? ? = 0, ? ? and ? (?) exist, ?(?) ?(?)=? (?) and ? (?) 0 then lim ? (?) ? ? To see this, note that ?(?) ?(?)= lim ? 0 What happens if ? ? = ? ? = 0? ? ? + ? ?(?)=? (?) ? ? + ? ?(?) ? ? + ? ?(?)= lim ? ? + ? ?(?) ? ? lim ? ? lim ? 0 ? (?) ? 0
LHpitals Rule L H pital s Rule: ? (?) ? (?) ?(?) ?(?)= lim If ? ? = ? ? = 0 then lim ? ? ? ? provided that the limit on the right exists and ? (?) 0 when ? is close to ? but not equal to it. L H pital s Rule for limits at infinity: Iflim ? ? ? = and lim lim ? ? right exists and ? (?) 0 when ? is sufficiently large. A similar statement holds for limits at . ? ? ? = then ? (?) ? (?)provided that the limit on the ?(?) ?(?)= lim
LHpitals Rule Examples 2?2+ 3? + 4 ?2 4? + 3 2? 4 2= 2 lim ? = lim ? = lim ? sin ? ? ?3 sin ? 6? cos ? 1 3?2 cos ? 6 lim ? 0 = lim ? 0 = 1 = lim ? 0 = lim ? 0 6
Objectives Know how to find the limit of sequences Know how to compute geometric series and telescoping series Corresponding sections in Simmons: 13.2, 13.3
Sequences If we have an element ??for every n, this is called a sequence Example: 1,2,3,4, ??= ? Example: 1,1 4, ??= 2,1 3,1 1 ?
Convergence of Sequences We say a sequence ?1,?2, converges to L if lim ? ??= ? Limits of sequences can be found in the same way as limits of functions. L Hopital s rule often works, sometimes other tools are needed. Example: If ??=?2+2? ? ??=1 2?2+5, lim ?2+ 2? ?, complete the ?2+ 2? ? = 1 2 ?+1 ? so lim 2 Example: If ??= square. ??= ?? ? + 1 ? + 12 1 ? ? ??= 1
Series A series is an expression of the form ?=1 ??=?1+ ?2+ ?3+ ?4+ Can be viewed as a sequence ??= ?=1 Key example: Geometric series Geometric series have the form 1 + ? + ?2+ ?3+ (times a constant) ? ??
Evaluating Series Geometric series can be evaluated as follows 1 + ? + + ?? 1=1 ?? 1 ? 1 ?? 1= ?=1 Partial fractions and cancelling terms can also be useful 1 ?if and only if ? < 1 1 1 ? 1 1 ? ?=2 1 ?= ?=1 1 When terms cancel like this, it is called a telescoping series. ?2+?= ?=1 ?+1= ?=1
