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INFINITE SERIESCALCULUS 3MATH148
Infinite Sequence-an unending succession of numbers, called terms, having a definite order.?1, ?2, ?3, ?4, …The formula or rule of the general term of a sequence, ?𝑛,is easily obtained since the sequence has a definite pattern that makes it easy to generate additional terms. For example,Sequence𝒂𝒏Brace Notation12,23,34,45, … ,?? + 1?? + 1𝑛 = 1+∞12,14,18,116, …12𝑛12𝑛𝑛 = 1+∞12, −23,34, −45, …−1𝑛+1?? + 1−1𝑛+1?? + 1𝑛 = 1+∞1, 3, 5, 7, …2? − 12? − 1𝑛 = 1+∞
Graphs of SequencesA sequence is a function whose domain is a set of integers. Typically, thedomain of a sequence is the set of positive numbers or set of nonnegativeintegers.Thegraphconsistsofasuccessionofisolatedpoints.Forexample, the graph of the sequence1𝑛𝑛 = 1+∞is the graph of the equation? =1𝑛, ? = 1, 2, 3, …and it is different from the graph of? =1𝑥, ? ≥ 1which is a continuous curve.
Limit of a SequenceSince a sequence?𝑛is only defined for integer values of?,the only limitthat makes sense is the limit of?𝑛as? → +∞.Informally speaking, the limitof a sequence?𝑛is intended to describe how?𝑛behaves as? → +∞.Asequence?
Illustrative ProblemsDetermine whether the series converges or diverges by examining the limit as ? → +∞.1.lim𝑛→+∞𝑛𝑛+12. lim𝑛→+∞[1 +−12𝑛]3. lim𝑛→∞𝑛2𝑛+14. lim𝑛→∞−1𝑛+1𝑛2𝑛+15. lim𝑛→∞−1𝑛+11?6. lim𝑛→∞8 − 2?7. lim𝑛→∞? + 18. lim𝑛→∞−1𝑛+1
Find the limit of the sequence 𝒆𝒏𝒏 = 𝟏+∞.lim𝑛→+∞??𝑛=(𝑖???????𝑖????)L’Hopital’sRule is applicable but cannot be applied directly to 𝑒𝑛,since the functions ?and 𝒏𝑛
𝑥
𝑛
Infinite Series
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Special Types of Series1. Geometric SeriesA geometric series 𝑘=0??