find the sum to infinity of the following series

rn
(1+1/n) -(1 - 1/1+n) - n square / (n+1)square + ...........

To find the sum to infinity of the given series, we will analyze each term and look for any patterns to determine if the series converges (has a finite sum) or diverges (has an infinite sum).

The series is as follows:
(1 + 1/n) - (1 - 1/(n+1)) - n^2 / (n+1)^2 + ...

Let's breakup the series into individual terms and investigate their behavior as n approaches infinity.

Term 1: 1 + 1/n
As n approaches infinity, 1/n approaches 0. Therefore, the term 1 + 1/n approaches 1.

Term 2: 1 - 1/(n+1)
Similar to Term 1, as n approaches infinity, 1/(n+1) approaches 0. Therefore, the term 1 - 1/(n+1) approaches 1.

Term 3: -n^2 / (n+1)^2
As n approaches infinity, both the numerator (-n^2) and denominator ((n+1)^2) approach infinity. However, we can simplify the term by dividing both the numerator and denominator by n^2, resulting in -1 / (1 + 1/n)^2. As shown earlier, we know that (1 + 1/n) approaches 1. Therefore, the term -1 / (1 + 1/n)^2 approaches -1 / 1^2, which simplifies to -1.

From the analysis, we can see that the series alternates between the values 1 and -1, regardless of the value of n. This implies that the terms do not approach a specific sum and the series diverges.

Therefore, the sum to infinity of the given series does not exist.