Consider the sequence whose n th term is given by the...

Consider the sequence whose $n$ th term is given by the following. In each case, determine the limit of the sequence by expressing the $n$ th term as a Riemann sum for a suitable function. (i) $\frac{1}{n^{17}} \sum_{i=1}^{n} i^{16}$, (ii) $\frac{1}{n^{5 / 2}} \sum_{i=1}^{n} i^{3 / 2}$, (iii) $\sum_{i=1}^{n} \frac{1}{\sqrt{i n+n^{2}}}$, (iv) $\frac{1}{n}\left\{\sum_{i=1}^{n}\left(\frac{i}{n}\right)+\sum_{i=n+1}^{2 n}\left(\frac{i}{n}\right)^{3 / 2}+\sum_{i=2 n+1}^{3 n}\left(\frac{i}{n}\right)^{2}\right\}$.

Consider the sequence whose $n$ th term is given by the following. In each case, determine the limit of the sequence by expressing the $n$ th term as a Riemann sum for a suitable function.
(i) $\frac{1}{n^{17}} \sum_{i=1}^{n} i^{16}$,
(ii) $\frac{1}{n^{5 / 2}} \sum_{i=1}^{n} i^{3 / 2}$,
(iii) $\sum_{i=1}^{n} \frac{1}{\sqrt{i n+n^{2}}}$,
(iv) $\frac{1}{n}\left\{\sum_{i=1}^{n}\left(\frac{i}{n}\right)+\sum_{i=n+1}^{2 n}\left(\frac{i}{n}\right)^{3 / 2}+\sum_{i=2 n+1}^{3 n}\left(\frac{i}{n}\right)^{2}\right\}$.

Key Concepts

Riemann Sum

A Riemann sum is a method for approximating the value of a definite integral by dividing the region under a curve into small sub-intervals, evaluating the function at specific points within these intervals, and then summing the products of these values with the widths of the intervals. This concept is central to transitioning from a discrete sum representation to its continuous integral form as the number of sub-intervals increases indefinitely.

Definite Integral

A definite integral calculates the accumulation of a quantity, such as area under a curve, over a specific interval. By taking the limit of a Riemann sum as the number of partitions tends to infinity, one obtains the exact value of the integral. This process connects discrete approximations with continuous analysis and is essential in evaluating the limits of sequences expressed as sums.

Change of Variables in Summation

Changing the variable from a discrete index, such as i in a sum, to a continuous variable, typically x = i/n, bridges the gap between a finite sum and an integral. This technique transforms the discrete terms in the sequence into a continuous function, thereby enabling the interpretation of the sum as a Riemann sum and facilitating the evaluation of the sequence’s limit via integration.

Limits of Sequences via Integrals

The limit of a sequence can often be determined by representing its general term as a Riemann sum that approximates a definite integral. As the number of terms increases indefinitely, the Riemann sum converges to the definite integral of a corresponding function. This approach links the behavior of sequences in the discrete setting to continuous integral calculus, providing a powerful tool for evaluation.

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