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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!