Sequences and Limiting Processes
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Combining limits
Equations are omitted for technical reasons - download the original pdf
1. Multiplying a function by a constant multiplies that limit by that constant. [Equation goes here - download the original to see it.] 2. The limit of a sum of sequences is the sum of the limits of the individual sequences. [Equation] Proof (1) If [Equation goes here - download the original to see it.] are null sequences, then so is [Equation] [Equation] [Equation] Therefore [Equation] is a null sequence. (2) If [Equation] then [Equation] are null sequences. Hence by (1) [Equation] is a null sequence. 3. The Limit of a product is the product of the limits. [Equation] Proof (1) If [Equation] is a null sequence and [Equation] is a bounded sequence, then [Equation] is a null sequence. Since [Equation] is bounded and [Equation] is a null sequence [Equation] Therefore [Equation] Therefore, [Equation] is a null sequence. (2) Suppose [Equation] We have [Equation] 4. The limit of a quotient is the quotient of the limits. [Equation] Proof (1) [Equation] [Equation] Then [Equation] [Equation] is a bounded sequence, hence [Equation] for some K. Hence [Equation] Hence [Equation] is a null sequence and [Equation] Example Evaluate [Equation] Solution[Equation]
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Contents of
Sequences and Limiting Processes
1 Sequences
2 Null sequences
3 Limits
4 Sequences tending to infinity
5 Alternating (or oscillating) sequence
6 Combining limits
7 Monotone sequences
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