Does ∑ n = 2 ∞ ( − 1 ) n ln ⁡ ( n ) have a closed form?

Question

Does       ∑          n      =      2              ∞                  (      −      1              )        n                    ln      ⁡      (      n      )       have a closed form?

stigliy0 · Accepted Answer

Consider the seriesThen      f    ′    (  x  )  =      ∑          n      =      1        ∞    (  −  1      )    n        n    x    +  1  =  1  −  η  (  −  x  )  =  1  +  (      2          1      +      x        −  1  )  ζ  (  −  x  )where   η and   ζ are the Dirchlet eta and Riemann zeta functions, respectively. Then the summation is given by  f  (  0  )  =      ∫          −      ∞        0    1  −  η  (  −  x  )    d  x  =      ∫    0    ∞    1  −  η  (  x  )    d  x  ≈  0.9243

Carolyn Beck · Accepted Answer

Use the polyogarithm      Li    s    ⁡  (  z  )  =      ∑          k      =      1        ∞              z      k              k      s        ,        |    z      |    &amp;lt;  1  .Compare this to            ∂      n              ∂              s        n                  Li    s    ⁡  (  z  )  =      ∑          k      =      1        ∞                      z        k            (      log      ⁡      k              )        n                    k      s                    ∂      n              ∂              s        n                  Li    s    ⁡  (  z  )                    |                    s      =      0        =      ∑          k      =      1        ∞        z    k    (  log  ⁡  k      )    n

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

Answered question

Answer & Explanation

New Questions in Calculus 2