Suppose that the nth partial sum of the series ∞
Σn=1 an is sn = 5n − 2 5n + 2
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To find the value of the series Σn=1 an, we need to find an expression for the terms an. Given that the nth partial sum is sn = 5n − 2 / (5n + 2), we can express an in terms of the partial sum by finding the difference between consecutive terms:
an = sn - sn-1
Substituting the given expression of sn, we have:
an = (5n − 2) / (5n + 2) - (5(n-1) − 2) / (5(n-1) + 2)
Simplifying this expression gives us:
an = (5n − 2) / (5n + 2) - (5n - 7) / (5n - 3)
Next, we can try to simplify this expression further by finding a common denominator. Multiplying the first term's denominator by (5n - 3) and the second term's denominator by (5n + 2) gives:
an = [(5n − 2)(5n - 3)] / [(5n + 2)(5n - 3)] - [(5n - 7)(5n + 2)] / [(5n + 2)(5n - 3)]
Simplifying this expression further gives us:
an = [25n^2 - 25n - 6n + 6] / [25n^2 - 9] - [25n^2 - 9n - 35n - 14] / [25n^2 - 9]
Combining like terms, we get:
an = (19n + 6) / (25n^2 - 9)
Now we have an expression for the terms an in the series. We can use this expression to find the value of the series by taking the limit as n approaches infinity:
Σn=1 an = lim(n→∞) (19n + 6) / (25n^2 - 9)
To evaluate this limit, we can apply the limit rules. The numerator grows linearly with n, while the denominator grows quadratically. Therefore, the dominant term in the denominator is n^2, and we can ignore the other terms in the denominator.
Taking the limit, we have:
Σn=1 an = lim(n→∞) (19n + 6) / (25n^2 - 9) = lim(n→∞) 19n / 25n^2 = lim(n→∞) 19/25n = 0
Therefore, the value of the series Σn=1 an is 0.