Evaluate : limn→∞ 1.2+2.3+3.4+...+n(n+1)n3
Question
Evaluate : limn→∞ 1.2+2.3+3.4+...+n(n+1)n3
Solution
To evaluate the given limit, we can rewrite the expression as a sum of fractions.
First, let's expand the expression:
1.2 + 2.3 + 3.4 + ... + n(n+1)/n^3
Next, we can simplify each term by dividing both the numerator and denominator by n^3:
(1/n^2) * (2/n) + (2/n^2) * (3/n) + (3/n^2) * (4/n) + ... + (n/n^2) * (n+1/n)
Now, we can rewrite each term as a fraction:
(2/n^3) + (6/n^3) + (12/n^3) + ... + (n(n+1)/n^3)
Next, we can factor out 1/n^3 from each term:
(1/n^3) * (2 + 6 + 12 + ... + n(n+1))
Now, let's simplify the sum of the terms inside the parentheses:
2 + 6 + 12 + ... + n(n+1) = n(n+1)(n+2)/2
Substituting this back into the expression, we have:
(1/n^3) * (n(n+1)(n+2)/2)
Now, we can simplify further by canceling out common factors:
(1/n^3) * (n(n+1)(n+2)/2) = (n+1)(n+2)/(2n^2)
Finally, as n approaches infinity, the terms (n+1) and (n+2) become insignificant compared to n^2. Therefore, we can ignore them in the limit calculation.
lim(n→∞) (n+1)(n+2)/(2n^2) = lim(n→∞) 1/2 = 1/2
So, the limit of the given expression as n approaches infinity is 1/2.
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