Calculus Examples

$lim_{n \to \infty} \sum_{i = 1}^{n} (\frac{2 i}{n}) (\frac{2}{n})$

Step 1

Simplify the summation.

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Step 1.1

Multiply $\frac{2 i}{n} \cdot \frac{2}{n}$ .

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Step 1.1.1

Multiply $\frac{2 i}{n}$ by $\frac{2}{n}$ .

$\frac{2 i \cdot 2}{n \cdot n}$

Step 1.1.2

Multiply $2$ by $2$ .

$\frac{4 i}{n \cdot n}$

Step 1.1.3

Raise $n$ to the power of $1$ .

$\frac{4 i}{n^{1} n}$

Step 1.1.4

Raise $n$ to the power of $1$ .

$\frac{4 i}{n^{1} n^{1}}$

Step 1.1.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 i}{n^{1 + 1}}$

Step 1.1.6

Add $1$ and $1$ .

$\frac{4 i}{n^{2}}$

Step 1.2

Rewrite the summation.

$\sum_{i = 1}^{n} \frac{4 i}{n^{2}}$

Step 2

The formula for the summation of a constant is:

$\sum_{k = 1}^{n} c = c n$

Step 3

Substitute the values into the formula.

$(\frac{4 i}{n^{2}}) (n)$

Step 4

Cancel the common factor of $n$ .

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Step 4.1

Factor $n$ out of $n^{2}$ .

$\frac{4 i}{n \cdot n} n$

Step 4.2

Cancel the common factor.

$\frac{4 i}{n \cdot n} n$

Step 4.3

Rewrite the expression.

$\frac{4 i}{n}$