Infinite series

[leaden geode]

integral representation?

[languid pine]

the first series is an alternating series that converges to a known value, specifically, it can be connected to the ditichlet eta function: integral(ln(1+x)/x,dx from 0 to 1) and
the second series is a alternating harmonic series for n terms. as n aproaches infinty , the infinite version of this converges to -ln(2) thus,there no direct integral representaion:

[leaden geode]

i found an intefral representation recently but i wonder if there are any other integrals

[leaden geode]

and from where do you even get to that conclusion: thus,there no direct integral representaion

this is the best i got fr an integral representation

and if you want to know the evaluation of the series is this