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# Calculus II - 22

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Stewart Calculus Section 11.2

Stewart Calculus Section 11.2

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• 1. 11.2 Infinite Series ∞Given a series == + + + ···let denotes its partial sum: = = + + ··· + =then { } is a new sequence. If it isconvergent: = ∞we call the series = convergent andwrite = . =
• 2. ∞Theorem: If the series = is convergent,then = .If = or does not exist, then ∞ = is divergent.On the contrary, if = we know ∞nothing about = .Two important examples: = = = =
• 3. = =Ex1 is an example of geometric series: = + + + + ··· = is called the common ratio.The geometric series = + + + + ··· =is convergent if | | < and its sum is = . =If | | the geometric series is divergent.
• 4. = =Ex2 is called harmonic series.
• 5. = =Ex2 is called harmonic series. ∞ = = + + + + + + + + + + + + + + + +··· > + + + + + + + + + + + + + + + +··· = + + + + +··· =∞
• 6. Ex3: = ( + )
• 7. Ex3: = ( + )Notice that = ( + ) +
• 8. Ex3: = ( + )Notice that = ( + ) +we have = = ( + ) = + = + + + ··· =
• 9. ∞ ∞Theorem: If = and = areconvergent, then ( + )= + = = = ( )= = = = ( )= = =Not true for product and quotient!