How can the sum of values be approximated for infinite intervals in java

Question

I am trying to write a piece of code in java that needs to calculate the sum of values in following manner for the range [-∞, +∞]

K= +∞

∑ [ f( (2K+1)*x ) - f( (2K-1)*x ) ]

K= -∞

There are two confusions that I am facing with the calculation to produce a computationally feasible and efficient solution:

1. what value must be assumed for ∞ to approximate the calculation as above?

2. what minimum value must be used to increment K within loop, assuming that K holds fractional values too.

Answer 1

First of all, in order to get an approximation you need the value to decrease dramatically. For example, -1/x. If x increases much the answer will be near zero.

1. What value must be assumed for ∞ to approximate the calculation as above?

for ( float k = 0 ; ; k++ ) 
{
    sum += (((2 * ( k) + 1) * x) - ((2 * ( k) - 1) * x));   //adding from 1 to infinity
    sum += (((2 * (-k) + 1) * x) - ((2 * (-k) - 1) * x));    //adding from -1 to infinity
}

but this will result in an infinite loop. You can replace x with 1/x and increase x each time in the loop. The code becomes

if ( (sum - lastsum) < 0.001 )
{
    break;
}
    lastsum = sum;
}

2. What minimum value must be used to increment K within [the] loop, assuming that K holds fractional values too?

This is according to the type of K. If it is double, it could have up to 15 decimal places.

In addition, in the summation, you should move by one, not a fraction. See https://upload.wikimedia.org/math/d/f/2/df26e1cf51b67fbedd01ce9c68cbbef5.png https://en.wikipedia.org/wiki/Summation