[sum_{n = 1}^infty {frac{1}{{{n^3}}}} left( {sumlimits_{k = 1}^n {frac{{{{left( { - 1}
ight)}^{k - 1}}}}{k}} }
ight) = frac{7}{4}zeta left( 3
ight)ln 2 - frac{{{pi ^4}}}{{288}}
]
[sum_{n = 1}^infty {frac{{{{left( { - 1}
ight)}^{n - 1}}}}{{{n^3}}}left( {sumlimits_{k = 1}^n {frac{{{{left( { - 1}
ight)}^{k - 1}}}}{k}} }
ight)} = frac{{{pi ^4}}}{{60}} + frac{{{pi ^2}}}{{12}}{ln ^2}2 - frac{1}{{12}}{ln ^4}2 - 2mathrm{Li}_4left( {frac{1}{2}}
ight)
]
[sum_{n = 1}^infty {frac{{{{left( { - 1}
ight)}^{n - 1}}}}{{{n^2}}}left( {sumlimits_{k = 1}^n {frac{1}{{{k^2}}}} }
ight)} = - frac{{17{pi ^4}}}{{480}} - frac{{{pi ^2}}}{6}{ln ^2}2 + frac{1}{6}{ln ^4}2 + 4mathrm{Li}_4left( {frac{1}{2}}
ight) + frac{7}{2}zeta left( 3
ight)ln 2
]
[{sum_{n = 1}^infty {frac{{{{left( { - 1}
ight)}^{n - 1}}}}{{{n^2}}}left( {sum_{k = 1}^n {frac{{{{left( { - 1}
ight)}^{k - 1}}}}{k}} }
ight)} ^2} = - frac{{61{pi ^4}}}{{1440}} + frac{{{pi ^2}}}{3}{ln ^2}2 + frac{1}{6}{ln ^4}2 + 4mathrm{Li}_4left( {frac{1}{2}}
ight) + frac{7}{4}zeta left( 3
ight)ln 2
]
[sum_{n = 1}^infty {frac{{{{left( { - 1}
ight)}^{n - 1}}}}{{{n^2}}}left( {sum_{k = 1}^n {frac{1}{k}} }
ight)left( {sum_{k = 1}^n {frac{{{{left( { - 1}
ight)}^{k - 1}}}}{k}} }
ight)} = frac{{29{pi ^4}}}{{1440}} + frac{{{pi ^2}}}{8}{ln ^2}2 - frac{1}{8}{ln ^4}2 - 3mathrm{Li}_4left( {frac{1}{2}}
ight)
]
[egin{align*}
sum_{n = 1}^infty {frac{{{{left( { - 1}
ight)}^{n - 1}}}}{{{n^3}}}{{left( {sum_{k = 1}^n {frac{1}{k}} }
ight)}^2}}&=- frac{1}{9}{pi ^2}{ln ^3}2 + frac{2}{{15}}{ln ^5}2 + 4mathrm{Li}_4left( {frac{1}{2}}
ight)ln 2 + 4mathrm{Li}_5left( {frac{1}{2}}
ight) \
&~~~- frac{{11}}{{48}}{pi ^2}zeta left( 3
ight) + frac{7}{2}zeta left( 3
ight){ln ^2}2 - frac{{19}}{{32}}zeta left( 5
ight)
end{align*}]
[egin{align*}
sum_{n = 1}^infty {frac{{{{left( { - 1}
ight)}^{n - 1}}}}{{{n^2}}}left( {sumlimits_{k = 1}^n {frac{1}{k}} }
ight)left( {sum_{k = 1}^n {frac{1}{{{k^2}}}} }
ight)}&= frac{1}{9}{pi ^2}{ln ^3}2 - frac{2}{{15}}{ln ^5}2 - 4mathrm{Li}_4left( {frac{1}{2}}
ight)ln 2 - 4mathrm{Li}_5left( {frac{1}{2}}
ight) \
&~~~+ frac{5}{{32}}{pi ^2}zeta left( 3
ight) - frac{7}{4}zeta left( 3
ight){ln ^2}2 + frac{{23}}{8}zeta left( 5
ight)
end{align*}]
[sum_{n = 1}^infty {frac{{{{left( { - 1}
ight)}^{n - 1}}}}{{{n^2}}}left( {sumlimits_{k = 1}^n {frac{1}{k}} }
ight)left( {sum_{k = 1}^n {frac{{{{left( { - 1}
ight)}^{k - 1}}}}{{{k^2}}}} }
ight)}= - frac{{13}}{{48}}{pi ^2}zeta left( 3
ight) + frac{{125}}{{32}}zeta left( 5
ight)
]
[egin{align*}
sum_{n = 1}^infty {frac{{{{left( { - 1}
ight)}^{n - 1}}}}{{{n^3}}}left( {sum_{k = 1}^n {frac{1}{k}} }
ight)left( {sum_{k = 1}^n {frac{{{{left( { - 1}
ight)}^{k - 1}}}}{k}} }
ight)}&= frac{2}{{45}}{pi ^4}ln 2 + frac{1}{{36}}{pi ^2}{ln ^3}2 - frac{1}{{60}}{ln ^5}2 + 2mathrm{Li}_5left( {frac{1}{2}}
ight) \
&~~~- frac{1}{{48}}{pi ^2}zeta left( 3
ight) - frac{7}{8}zeta left( 3
ight){ln ^2}2 - frac{{37}}{{16}}zeta left( 5
ight)
end{align*}]
[egin{align*}
sum_{n = 1}^infty {frac{1}{{{n^3}}}left( {sumlimits_{k = 1}^n {frac{1}{k}} }
ight)left( {sumlimits_{k = 1}^n {frac{{{{left( { - 1}
ight)}^{k - 1}}}}{k}} }
ight)} &= frac{2}{{45}}{pi ^4}ln 2 + frac{1}{{36}}{pi ^2}{ln ^3}2 - frac{1}{{60}}{ln ^5}2 + 2mathrm{Li}_5left( {frac{1}{2}}
ight) \
&~~~+ frac{1}{{16}}{pi ^2}zeta left( 3
ight) - frac{7}{8}zeta left( 3
ight){ln ^2}2 - frac{{193}}{{64}}zeta left( 5
ight)
end{align*}]
[egin{align*}
sum_{n = 1}^infty {frac{{{{left( { - 1}
ight)}^{n - 1}}}}{{{n^2}}}{{left( {sumlimits_{k = 1}^n {frac{1}{k}} }
ight)}^3}} &= - frac{1}{6}{pi ^2}{ln ^3}2 + frac{1}{5}{ln ^5}2 + 6mathrm{Li}_4left( {frac{1}{2}}
ight)ln 2 + 6mathrm{Li}_5left( {frac{1}{2}}
ight) \
&~~~- frac{9}{{32}}{pi ^2}zeta left( 3
ight) + frac{{21}}{8}zeta left( 3
ight){ln ^2}2 - frac{9}{4}zeta left( 5
ight)
end{align*}]
[egin{align*}
sum_{n = 1}^infty {frac{1}{{{n^2}}}{{left( {sumlimits_{k = 1}^n {frac{{{{left( { - 1}
ight)}^{k - 1}}}}{k}} }
ight)}^3}} &= - frac{{29}}{{160}}{pi ^4}ln 2 + frac{{11}}{{12}}{pi ^2}{ln ^3}2 - frac{1}{{20}}{ln ^5}2 - 6mathrm{Li}_4left( {frac{1}{2}}
ight)ln 2 \
&~~~- 24mathrm{Li}_5left( {frac{1}{2}}
ight) + frac{1}{{12}}{pi ^2}zeta left( 3
ight) + frac{{367}}{{16}}zeta left( 5
ight)
end{align*}]
[egin{align*}
&sum_{n = 1}^infty {frac{{{{left( { - 1}
ight)}^{n - 1}}}}{{{n^2}}}left( {sum_{k = 1}^n {frac{{{{left( { - 1}
ight)}^{k - 1}}}}{k}} }
ight)left( {sum_{k = 1}^n {frac{{{{left( { - 1}
ight)}^{k - 1}}}}{{{k^2}}}} }
ight)} \&= frac{{49}}{{720}}{pi ^4}ln 2 - frac{1}{{18}}{pi ^2}{ln ^3}2 + frac{1}{{10}}{ln ^5}2 + 4mathrm{Li}_4left( {frac{1}{2}}
ight)ln 2+ 8mathrm{Li}_5left( {frac{1}{2}}
ight) + frac{1}{{96}}{pi ^2}zeta left( 3
ight) - frac{{35}}{4}zeta left( 5
ight)
end{align*}]
[sum_{n = 1}^infty {frac{1}{{{n^3}}}left( {sum_{k = 1}^n {frac{1}{k}} }
ight)left( {sum_{k = 1}^n {frac{1}{{{k^2}}}} }
ight)} = - frac{{101}}{{45360}}{pi ^6} + frac{5}{2}{zeta ^2}left( 3
ight)
]
[sum_{n = 1}^infty {frac{1}{{{n^3}}}{{left( {sumlimits_{k = 1}^n {frac{1}{k}} }
ight)}^3} = } frac{{31}}{{5040}}{pi ^6} - frac{5}{2}{zeta ^2}left( 3
ight)
]