Calculus Examples

lim n \to \infty \frac{1}{n} n \sum i = 1 \frac{1}{1 + {(\frac{i}{n})}^{2}}

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

Step 1

Simplify the summation.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Apply the product rule to

\frac{i}{n}

$\frac{i}{n}$ .

\frac{1}{1 + \frac{i^{2}}{n^{2}}}

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

Step 1.1.2

Write

1

$1$ as a fraction with a common denominator.

\frac{1}{\frac{n^{2}}{n^{2}} + \frac{i^{2}}{n^{2}}}

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

Step 1.1.3

Combine the numerators over the common denominator.

\frac{1}{\frac{n^{2} + i^{2}}{n^{2}}}

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

\frac{1}{\frac{n^{2} + i^{2}}{n^{2}}}

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

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

1 \frac{n^{2}}{n^{2} + i^{2}}

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

Step 1.3

Multiply

\frac{n^{2}}{n^{2} + i^{2}}

$\frac{n^{2}}{n^{2} + i^{2}}$ by

1

$1$ .

\frac{n^{2}}{n^{2} + i^{2}}

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

Step 1.4

Rewrite the summation.

\frac{1}{n n \sum i = 1 \frac{n^{2}}{n^{2} + i^{2}}}

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

\frac{1}{n n \sum i = 1 \frac{n^{2}}{n^{2} + i^{2}}}

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

Step 2

The formula for the summation of a constant is:

n \sum k = 1 c = c n

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

Step 3

Substitute the values into the formula and make sure to multiply by the front term.

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

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

Step 4

Simplify.

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Step 4.1

Cancel the common factor of

n

$n$ .

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Step 4.1.1

Factor

n

$n$ out of

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

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

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Combine

$\frac{n}{n^{2} + i^{2}}$ and

$n$ .

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

Step 4.3

Raise

$n$ to the power of

$1$ .

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

Step 4.4

Raise

$n$ to the power of

$1$ .

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

Step 4.5

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.6

Add

$1$ and

$1$ .

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

Step 4.7

Simplify the denominator.

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Step 4.7.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{n^{2}}{n^{2} - 1}$

Step 4.7.2

Rewrite

$n^{2} - 1$ in a factored form.

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Step 4.7.2.1

Rewrite

$1$ as

$1^{2}$ .

$\frac{n^{2}}{n^{2} - 1^{2}}$

Step 4.7.2.2

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = n$ and

$b = 1$ .

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