Does the series ∑ n = 1 ∞ log ⁡ ( n + 1 n ) converge?

Question

Does the series       ∑          n      =      1              ∞        log  ⁡  (            n      +      1        n    ) converge?

Govorei9b · Accepted Answer

Your argument that       b    n    →  0 can be made much simpler:             n      +      1        n    →  1 and since   log is continuous so   log  ⁡  (            n      +      1        n    )  →  log  ⁡  1  =  0. However this won&#039;t help as your proof breaks down after that: you say that there is some convergent   ∑      a    n   such that       a    n    →  L  ≠  0 but this is not possible as if   ∑      a    n   converges then       a    n    →  0We know that   log  ⁡  x  ∼  x  −  1 for   x  →  1 (as follows from the Taylor series expansion of   log  ⁡  x). Since             k      +      1        k    →  1 this means that your series   ∑      b    n   behaves roughly like   ∑  (            k      +      1        k    −  1  )  =  ∑      1    k   and hence diverges. To prove this rigorously use the comparison test with       ∑          n      =      1              ∞            1    n  Another way to prove divergence is by writing the partial sums of the series and noticing that by telescoping all elements cancel and you are left with       ∑          k      =      1        n    log  ⁡  (            k      +      1        k    )  =  log  ⁡  (  n  +  1  )  →  ∞

Does the series <munderover> ∑ <mrow class="MJX-TeXAtom-ORD"> n

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