Find a formula for the sum of n terms. Use the formula to...

Find a formula for the sum of $n$ terms. Use the formula to find the limit as $n \rightarrow \infty.$
$$\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(1+\frac{2 i}{n}\right)^{2}\left(\frac{2}{n}\right)$$

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Key Concepts

Riemann Sum

A Riemann sum is a method for approximating the area under a curve by dividing the interval into small subintervals, selecting sample points in each subinterval, and summing up the products of the function values at these sample points and the widths of the subintervals. This technique provides a bridge between discrete sums and continuous integrals.

Definite Integral

The definite integral is a fundamental concept in calculus that represents the exact area under a curve between two points. It is formally defined as the limit of a Riemann sum as the number of subintervals approaches infinity, allowing for the precise calculation of accumulated quantities in continuous contexts.

Limit Process

The limit process is crucial in calculus, particularly when transitioning from a discrete sum to a continuous integral. It involves evaluating the behavior of a sequence or function as the number of terms or subdivisions increases indefinitely, ensuring that the approximation (like a Riemann sum) converges to an exact value (the definite integral).

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