Sum of infinite arithmetico geometric sequences

The sum of 'n' terms of an arithmetico geometric sequences was derived in the last post.

The sum of infinite number of terms of an arithmetico geometric sequence is derived from the formula obtained in the last post. We had derived the following formula:

Sum of 'n' terms of an arithmetico geometric sequence = 

Note that we can only apply the below procedure if the value of r is between 1 and -1, otherwise there is no method to find the formula for sum of infinite arithmetico geometric sequence. That is, the following condition should be met:

| r | < 1

Now, if an arithmetico geometric sequence is infinite, it means that the value of 'n' in the above formula is equal to infinity. That is,

Now, in the above formula, as 'n' nears infinite, so the the value of r^n-1 and rⁿ nears zero. This is because if a number lesser than 1 is multiplied with itself, a smaller number is obtained. Then, if it is again multiplied with itself a still smaller number is obtained. Further, as its exponent nears infinity (that is n and (n - 1)), then the number is multiplied with itself infinite number of times, resulting in a value almost equal to zero.

We can represent this mathematically as follows:

As so rⁿ → 0 and r^{n - 1} → 0

Now keeping in mind that in the above formula rⁿ and r^{n - 1} both equal zero, we can rewrite the formula as:

This formula is for the sum of infinite arithmetico geometric sequence.