Calculus Examples

$\sum_{0}^{\infty} \frac{2 n^{2} - n^{3}}{2 n^{3} + 5}$

Step 1

The series is divergent if the limit of the sequence as $n$ approaches $\infty$ does not exist or is not equal to $0$ .

$lim_{n \to \infty} \frac{2 n^{2} - n^{3}}{2 n^{3} + 5}$

Step 2

Evaluate the limit.

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Step 2.1

Divide the numerator and denominator by the highest power of $n$ in the denominator, which is $n^{3}$ .

$lim_{n \to \infty} \frac{\frac{2 n^{2}}{n^{3}} - \frac{n^{3}}{n^{3}}}{\frac{2 n^{3}}{n^{3}} + \frac{5}{n^{3}}}$

Step 2.2

Evaluate the limit.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $n^{2}$ and $n^{3}$ .

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Step 2.2.1.1.1

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

$lim_{n \to \infty} \frac{\frac{n^{2} \cdot 2}{n^{3}} - \frac{n^{3}}{n^{3}}}{\frac{2 n^{3}}{n^{3}} + \frac{5}{n^{3}}}$

Step 2.2.1.1.2

Cancel the common factors.

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Step 2.2.1.1.2.1

Factor $n^{2}$ out of $n^{3}$ .

$lim_{n \to \infty} \frac{\frac{n^{2} \cdot 2}{n^{2} n} - \frac{n^{3}}{n^{3}}}{\frac{2 n^{3}}{n^{3}} + \frac{5}{n^{3}}}$

Step 2.2.1.1.2.2

Cancel the common factor.

$lim_{n \to \infty} \frac{\frac{n^{2} \cdot 2}{n^{2} n} - \frac{n^{3}}{n^{3}}}{\frac{2 n^{3}}{n^{3}} + \frac{5}{n^{3}}}$

Step 2.2.1.1.2.3

Rewrite the expression.

$lim_{n \to \infty} \frac{\frac{2}{n} - \frac{n^{3}}{n^{3}}}{\frac{2 n^{3}}{n^{3}} + \frac{5}{n^{3}}}$

Step 2.2.1.2

Cancel the common factor of $n^{3}$ .

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Step 2.2.1.2.1

Cancel the common factor.

$lim_{n \to \infty} \frac{\frac{2}{n} - \frac{n^{3}}{n^{3}}}{\frac{2 n^{3}}{n^{3}} + \frac{5}{n^{3}}}$

Step 2.2.1.2.2

Rewrite the expression.

$lim_{n \to \infty} \frac{\frac{2}{n} - 1 \cdot 1}{\frac{2 n^{3}}{n^{3}} + \frac{5}{n^{3}}}$

Step 2.2.1.3

Multiply $- 1$ by $1$ .

$lim_{n \to \infty} \frac{\frac{2}{n} - 1}{\frac{2 n^{3}}{n^{3}} + \frac{5}{n^{3}}}$

Step 2.2.2

Cancel the common factor of $n^{3}$ .

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Step 2.2.2.1

Cancel the common factor.

$lim_{n \to \infty} \frac{\frac{2}{n} - 1}{\frac{2 n^{3}}{n^{3}} + \frac{5}{n^{3}}}$

Step 2.2.2.2

Divide $2$ by $1$ .

$lim_{n \to \infty} \frac{\frac{2}{n} - 1}{2 + \frac{5}{n^{3}}}$

Step 2.2.3

Split the limit using the Limits Quotient Rule on the limit as $n$ approaches $\infty$ .

$\frac{lim_{n \to \infty} \frac{2}{n} - 1}{lim_{n \to \infty} 2 + \frac{5}{n^{3}}}$

Step 2.2.4

Split the limit using the Sum of Limits Rule on the limit as $n$ approaches $\infty$ .

$\frac{lim_{n \to \infty} \frac{2}{n} - lim_{n \to \infty} 1}{lim_{n \to \infty} 2 + \frac{5}{n^{3}}}$

Step 2.2.5

Move the term $2$ outside of the limit because it is constant with respect to $n$ .

$\frac{2 lim_{n \to \infty} \frac{1}{n} - lim_{n \to \infty} 1}{lim_{n \to \infty} 2 + \frac{5}{n^{3}}}$

Step 2.3

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{n}$ approaches $0$ .

$\frac{2 \cdot 0 - lim_{n \to \infty} 1}{lim_{n \to \infty} 2 + \frac{5}{n^{3}}}$

Step 2.4

Evaluate the limit.

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Step 2.4.1

Evaluate the limit of $1$ which is constant as $n$ approaches $\infty$ .

$\frac{2 \cdot 0 - 1 \cdot 1}{lim_{n \to \infty} 2 + \frac{5}{n^{3}}}$

Step 2.4.2

Split the limit using the Sum of Limits Rule on the limit as $n$ approaches $\infty$ .

$\frac{2 \cdot 0 - 1 \cdot 1}{lim_{n \to \infty} 2 + lim_{n \to \infty} \frac{5}{n^{3}}}$

Step 2.4.3

Evaluate the limit of $2$ which is constant as $n$ approaches $\infty$ .

$\frac{2 \cdot 0 - 1 \cdot 1}{2 + lim_{n \to \infty} \frac{5}{n^{3}}}$

Step 2.4.4

Move the term $5$ outside of the limit because it is constant with respect to $n$ .

$\frac{2 \cdot 0 - 1 \cdot 1}{2 + 5 lim_{n \to \infty} \frac{1}{n^{3}}}$

Step 2.5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{n^{3}}$ approaches $0$ .

$\frac{2 \cdot 0 - 1 \cdot 1}{2 + 5 \cdot 0}$

Step 2.6

Simplify the answer.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Multiply $2$ by $0$ .

$\frac{0 - 1 \cdot 1}{2 + 5 \cdot 0}$

Step 2.6.1.2

Multiply $- 1$ by $1$ .

$\frac{0 - 1}{2 + 5 \cdot 0}$

Step 2.6.1.3

Subtract $1$ from $0$ .

$\frac{- 1}{2 + 5 \cdot 0}$

Step 2.6.2

Simplify the denominator.

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Step 2.6.2.1

Multiply $5$ by $0$ .

$\frac{- 1}{2 + 0}$

Step 2.6.2.2

Add $2$ and $0$ .

$\frac{- 1}{2}$

Step 2.6.3

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 3

The limit exists and does not equal $0$ , so the series is divergent.

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