Translating math to JavaScript; convergent/divergent series

Question

I am trying calculate the following series in JavaScript:

My code is as follows:

var res = []
for(var i=0; i<1000; i++) {
    res.push(i / ((i * Math.sqrt(i + 1)) + ((i + 1) * Math.sqrt(i))))
}

But this makes the series possibly converge towards 0 rather than approach 1. Here's the first 150 steps:

Is there something wrong with my translation from math to JavaScript? Maybe my parentheses?

UPDATE

As per @Barmar 's answer, the correct code shows convergence to 1 only for small values of infinity, diverging after 4 steps:

You're pushing the series into an array, not adding them together. — Barmar, Mar 12 '21 at 20:24
Sorry removed index. In my original code they were all index. — Cybernetic, Mar 12 '21 at 20:25
If you add up all the array elements, you'll get something approaching 1. — Barmar, Mar 12 '21 at 20:25
If your loop starts at 0, the first term is `0 / 0`, which is `NaN`. — ASDFGerte, Mar 12 '21 at 20:26
According to [wolfram](https://www.wolframalpha.com/input/?i=sum+from+k%3D1+to+infinity+k+%2F+%28k+*+sqrt%28k+%2B+1%29+%2B+%28k+%2B+1%29+*+sqrt%28k%29%29), the series diverges (which is btw also what i see in javascript). You seem to be plotting individual terms, instead of the sum. — ASDFGerte, Mar 12 '21 at 20:29
Oops, yes plotting the individual terms instead of sum. Thank you. — Cybernetic, Mar 12 '21 at 20:34

Barmar · Accepted Answer · 2021-03-12T20:49:08.447

1

You need to add each element of the series to the previous sum.

You also need to start at i = 1 to avoid dividing by 0.

console.config({
  maxEntries: Infinity
});

var res = [];
var sum = 0;
for (var i = 1; i < 1000; i++) {
  sum += i / ((i * Math.sqrt(i + 1)) + ((i + 1) * Math.sqrt(i)));
  res.push(sum);
}
console.log(res);

.as-console-wrapper {
  max-height: 100% !important;
}

Note that this shows that the series doesn't actually converge to 1. Which is also apparent from your incorrect graph, since you can see that the first 5 elements are between 0.195 and 0.293, and these add up to more than 1.

edited Mar 12 '21 at 20:49

answered Mar 12 '21 at 20:36

Barmar

741,623
53
500
612

Right, the conjecture he's trying to prove is apparently false. – Barmar Mar 12 '21 at 20:44
Every term is positive, so the moment you see a partial sum larger than one, it can't be true. – ASDFGerte Mar 12 '21 at 20:45
Correct, the series hits 1 at 4 steps, then diverges from there. – Cybernetic Mar 12 '21 at 20:45
3

In other words, it converges to 1 for small values of infinity. – Barmar Mar 12 '21 at 20:46

Translating math to JavaScript; convergent/divergent series

1 Answers1