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Series - Series The harmonic series is given by the...

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Series The harmonic series is given by the following formula 1 n n = 1 ! " . Its limit is 1 n n = 1 ! " = ! . There are several clear proofs of this property. One that is especially simple is that if the series does not go to infinity, then the series is bounded and must have a limit. In that case denote the limit by 1 n n = 1 ! " = # < ! . Then we must have that 1 2 n n = 1 ! " = # 2 . It should be obvious that 1 2 n ! 1 n = 1 " # = 1 + 1 3 + 1 5 + ! > $ 2 . However, this implies that 1 2 n n = 1 ! " + 1 2 n # 1 = 1 2 + 1 4 + 1 6 + ! + 1 + 1 3 + 1 5 + 1 7 + ! > $ 2 + $ 2 = $ n = 1 ! " . The problem with this is that 1 2 n n = 1 ! " + 1 2 n # 1 = 1 n n = 1 ! " = $ n = 1 ! " and this implies that ! > ! , a contradiction. Another elementary proof that 1 n n = 1 ! " = ! is given by the following observation. 1 n n = 2 k 2 k + 1 ! 1 " > 1 2 k + 1 n = 2 k 2 k + 1 ! 1 " = 2 k 2 k + 1 = 1 2 . This implies that 1 n n = 1 ! " > 1 2 + 1 2 + 1 2 + ! = ! . The last proof uses integration. That is beyond this part of the course, but it is helpful to see how it is used. It is easy to see that 1 x dx 1 N + 1 ! < 1 n n = 1 N " by comparing areas in the following diagram.
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