How to accumulate in the image collection in Google Earth Engine

Question

Here is an image collection [img1,img2,img3,img4,img5]. I want to get List or image collection [img1, img1+img2,img1+img2+img3, img1+img2+img3+img4, img1+img2+img3+img4+img5].

I tried to complete the following code https://code.earthengine.google.com/20763e4a37f2341558c2b7162f02eb83. But the result image collection b did not contain 'system:time_start'. I don't know how to modify my algorithm. Or you have a better way to get the accumulated image collection with 'system:time_start'.

var img1 = ee.Image(1).set('system:time_start','2001-01-01');
var img2 = ee.Image(2).set('system:time_start','2001-01-02');
var img3 = ee.Image(3).set('system:time_start','2001-01-03');
var img4 = ee.Image(4).set('system:time_start','2001-01-04');
var img5 = ee.Image(5).set('system:time_start','2001-01-05');
var image=ee.ImageCollection([img1,img2,img3,img4,img5]);
/*image = image.map(function(image) {
  return image.add(1);
});*/
var list = image.toList(image.size());

var t = ee.List(image.map(function(tt){
  return tt.get('system:time_start');
}))

var newList = list.slice(1);
newList = newList.iterate(function(data, list) {
  data = ee.Image(data);
  list = ee.List(list);
  list = list.add(data.add(ee.Image(list.get(list.size().subtract(1)))));
  return list;
}, ee.List([list.get(0)]));
var b = ee.ImageCollection(ee.List(newList).slice(1));


print(b) 

Answer 1

Following script creates a new collection with timestamp (collection_ts) from your old named variable image (now is collection).

var img1 = ee.Image(1).set('system:time_start','2001-01-01');
var img2 = ee.Image(2).set('system:time_start','2001-01-02');
var img3 = ee.Image(3).set('system:time_start','2001-01-03');
var img4 = ee.Image(4).set('system:time_start','2001-01-04');
var img5 = ee.Image(5).set('system:time_start','2001-01-05');

var collection = ee.ImageCollection([img1,img2,img3,img4,img5]); //this is a collection

var collection_ts = collection.map(function(img){
  var date = img.get('system:time_start');
  return img.set('system_time_start', date);
});

print(collection_ts);

var list = collection_ts.toList(collection_ts.size());

var newList = list.slice(1);

newList = newList.iterate(function(data, list) {
  data = ee.Image(data);
  list = ee.List(list);
  list = list.add(data.add(ee.Image(list.get(list.size().subtract(1)))));
  return list;
}, ee.List([list.get(0)]));

var b = ee.ImageCollection(ee.List(newList).slice(1));

print(b);

When you print collection_ts, it can be observed in Console Tab that it contains effectively a timestamp: