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Уральский Федеральный университет КВИВАЛЕНТНЫЕ БЕСКОНЕЧНО МАЛЫЕ ФУНКЦИИ Сравне...

Oct 18, 2024

Solution by Steps

step 1

We start with the limit:

\lim_{n \to \infty} \sum_{k=1}^{n} \frac{(3 + \frac{k}{n})^4}{n} = \frac{781}{5}

step 2

Next, we evaluate:

\lim_{n \to \infty} \sum_{k=1}^{n} \frac{(3 + \frac{k}{n})^4 \cdot 2}{n} = \frac{1562}{5}

step 3

Then, we consider:

\lim_{n \to \infty} \sum_{k=1}^{n} \frac{(3 + \frac{2k}{n})^4}{n} = \frac{1441}{5}

step 4

Finally, we find:

\lim_{n \to \infty} \sum_{k=1}^{n} \frac{(3 + \frac{2k}{n})^4 \cdot 2}{n} = \frac{2882}{5}

Answer

The results of the limits are:

\frac{781}{5}

\frac{1562}{5}

\frac{1441}{5}

, and

\frac{2882}{5}

respectively.

Key Concept

Understanding limits and sums of functions as

n

approaches infinity is crucial in calculus.

Explanation

The limits calculated represent the behavior of the sums of polynomial functions as the number of terms increases indefinitely, showcasing the convergence to specific values.