If the nth partial sum of the series ∞∑n=1 an is s_n=√(4n^2+5n−7)/(5n+10). Find the sum of the series

To find the sum of the series, we need to determine the limit of the sequence of partial sums as n approaches infinity.

The nth partial sum of the series is denoted as s_n, given by the formula:

s_n = √((4n^2 + 5n - 7) / (5n + 10))

To find the sum of the series, we need to determine the limit of s_n as n approaches infinity. Let's proceed step by step:

1. First, let's simplify the expression inside the square root. Multiply both the numerator and the denominator by 1/n:

s_n = √((4n^2 + 5n - 7) / (5n + 10)) * (√((1/n) / (1/n)))

2. Simplify the expression:

s_n = √(((4n^2 + 5n - 7) / (n * (5n + 10)))) = √(((4 + 5/n - 7/n^2) / (5 + 10/n)))

3. Take the limit of s_n as n approaches infinity:

lim(n→∞) √(((4 + 5/n - 7/n^2) / (5 + 10/n)))

As n approaches infinity, terms with 1/n and 1/n^2 become increasingly negligible. Thus:

lim(n→∞) √(4/5)

4. Evaluate the limit:

lim(n→∞) √(4/5) = √(4/5) = 2/√5

Therefore, the sum of the series is 2/√5.

√(4n^2+5n−7)/(5n+10) -> 2n/5n = 2/5