You can use Accumulate on example1 to get the same table produced by Table[Sum[N[x^(x/r), 5], {r, 1, k}],{k,1,10},{x,1,10}]:
example3 = Accumulate[example1]
{{1.0000, 4.0000, 27.000, 256.00, 3125.0, 46656., 8.235410^5, 1.677710^7, 3.874210^8, 1.000010^10},
{2.0000, 6.0000, 32.196, 272.00, 3180.9, 46872., 8.244510^5, 1.678110^7, 3.874410^8, 1.000010^10},
{3.0000, 7.5874, 35.196, 278.35, 3195.5, 46908., 8.245410^5, 1.678210^7, 3.874410^8, 1.000010^10},
{4.0000, 9.0016, 37.476, 282.35, 3203.0, 46923., 8.245710^5, 1.678210^7, 3.874410^8, 1.000010^10},
{5.0000, 10.321, 39.409, 285.38, 3208.0, 46931., 8.245910^5, 1.678210^7, 3.874410^8, 1.000010^10},
{6.0000, 11.581, 41.141, 287.90, 3211.8, 46937., 8.246010^5, 1.678210^7, 3.874410^8, 1.000010^10},
{7.0000, 12.800, 42.742, 290.11, 3215.0, 46942., 8.246110^5, 1.678210^7, 3.874410^8, 1.000010^10},
{8.0000, 13.989, 44.252, 292.11, 3217.7, 46946., 8.246110^5, 1.678210^7, 3.874410^8, 1.000010^10},
{9.0000, 15.156, 45.694, 293.96, 3220.2, 46949., 8.246210^5, 1.678210^7, 3.874410^8, 1.000010^10},
{10.000, 16.304, 47.085, 295.70, 3222.4, 46952., 8.246210^5, 1.678210^7, 3.874410^8, 1.000010^10}}
%example3 == Table[Sum[N[x^(x/r), 5], {r, 1, k}],{k,1,10},{x,1,10}]
True