You can use [`Accumulate`](http://reference.wolfram.com/language/ref/Accumulate.html) 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., 8.2354*10^5, 
  1.6777*10^7, 3.8742*10^8, 1.0000*10^10},  
 {2.0000, 6.0000, 32.196, 
  272.00, 3180.9, 46872., 8.2445*10^5, 1.6781*10^7, 3.8744*10^8, 
  1.0000*10^10},   
{3.0000, 7.5874, 35.196, 278.35, 3195.5, 46908., 
  8.2454*10^5, 1.6782*10^7, 3.8744*10^8, 1.0000*10^10},  
 {4.0000, 
  9.0016, 37.476, 282.35, 3203.0, 46923., 8.2457*10^5, 1.6782*10^7, 
  3.8744*10^8, 1.0000*10^10},   
{5.0000, 10.321, 39.409, 285.38, 3208.0,
   46931., 8.2459*10^5, 1.6782*10^7, 3.8744*10^8, 
  1.0000*10^10},   
{6.0000, 11.581, 41.141, 287.90, 3211.8, 46937., 
  8.2460*10^5, 1.6782*10^7, 3.8744*10^8, 1.0000*10^10},  
 {7.0000, 
  12.800, 42.742, 290.11, 3215.0, 46942., 8.2461*10^5, 1.6782*10^7, 
  3.8744*10^8, 1.0000*10^10},  
 {8.0000, 13.989, 44.252, 292.11, 3217.7,
   46946., 8.2461*10^5, 1.6782*10^7, 3.8744*10^8, 
  1.0000*10^10},  
 {9.0000, 15.156, 45.694, 293.96, 3220.2, 46949., 
  8.2462*10^5, 1.6782*10^7, 3.8744*10^8, 1.0000*10^10},   
{10.000, 
  16.304, 47.085, 295.70, 3222.4, 46952., 8.2462*10^5, 1.6782*10^7, 
  3.8744*10^8, 1.0000*10^10}}

    example3  == Table[Sum[N[x^(x/r), 5], {r, 1, k}],{k,1,10},{x,1,10}]
> True