Calculus Examples

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

Step 1

Simplify the summation.

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Step 1.1

Multiply $1 + \frac{i}{n}$ by $\frac{2}{n}$ .

$\frac{(1 + \frac{i}{n}) \cdot 2}{n}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

$\frac{(\frac{n}{n} + \frac{i}{n}) \cdot 2}{n}$

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 1.3

Combine $\frac{n + i}{n}$ and $2$ .

$\frac{\frac{(n + i) \cdot 2}{n}}{n}$

Step 1.4

Move $2$ to the left of $n + i$ .

$\frac{\frac{2 (n + i)}{n}}{n}$

Step 1.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 (n + i)}{n} \cdot \frac{1}{n}$

Step 1.6

Multiply $\frac{2 (n + i)}{n} \cdot \frac{1}{n}$ .

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Step 1.6.1

Multiply $\frac{2 (n + i)}{n}$ by $\frac{1}{n}$ .

$\frac{2 (n + i)}{n \cdot n}$

Step 1.6.2

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

$\frac{2 (n + i)}{n^{1} n}$

Step 1.6.3

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

$\frac{2 (n + i)}{n^{1} n^{1}}$

Step 1.6.4

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

$\frac{2 (n + i)}{n^{1 + 1}}$

Step 1.6.5

Add $1$ and $1$ .

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

Step 1.7

Rewrite the summation.

$\sum_{i = 1}^{n} \frac{2 (n + i)}{n^{2}}$

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.

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

Step 4

Cancel the common factor of $n$ .

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Step 4.1

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

$\frac{2 (n + i)}{n \cdot n} n$

Step 4.2

Cancel the common factor.

$\frac{2 (n + i)}{n \cdot n} n$

Step 4.3

Rewrite the expression.

$\frac{2 (n + i)}{n}$